improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy 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 grunwald 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

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

the structure and analysis of spherical time dependent processes the class of all spherical processes is investigated irrespective of moments properties of these processes are studied including their structure representations and linear and nonlinear spaces a basic representation is x t a 1 2 y t t in t where ais nonnegative and y t is gaussian time series analysesnot requiring moments are set forth in both the time and frequency domains including spectral analysis solutions of forecasting problems are found to be linear when the process is spherical with the property that errors of the optimal forecasts are stochasticalay most concentrated about zero these developments apply to the alpha sub gaussian processes and are advanced for use with certain heavy tailed processes arising in economics engineering and the physical sciences

linear systems with levy fluctuations abstract we have examined the response of multivariable linear systems to levy fluctuations the characteristic function of the response is obtained explicitly as a function of time using the characteristic function the fractional moments of the process are determined the nature of the product rule obeyed by the characteristic function of a general linear dynamical system is elucidated

diffusion in heterogeneous media diffusion processes in heterogeneous media and biological systems in particular are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory or a special combination of the two accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well this problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time dependent coefficients we propose an iterative scheme for solving fractional differential equations with time dependent coefficients that is based on a parametric expansion in the fractional index we demonstrate how this method can be used to predict the long time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources

theory of adiabatic fluctuations third order noise we consider the response of a dynamical system driven by external adiabatic fluctuations based on the adiabatic following approximation we have made a systematic separation of timescales to carry out an expansion in where is the strength of fluctuations and is the damping rate we show that the probability distribution functions obey the differential equations of motion which contain third order terms beyond the usual fokker planck terms leading to non gaussian noise the problem of adiabatic fluctuations in velocity space which is the counterpart of brownian motion for fast fluctuations has been solved exactly the characteristic function and the associated probability distribution function are shown to be of stable form the linear dissipation leads to a steady state which is stable and the variances and higher moments are shown to be finite

fractional diffusion probability distributions and random walk models abstract we present a variety of models of random walk discrete in space and time suitable for simulating random variables whose probability density obeys a space time fractional diffusion equation

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

analogies between colored levy noise and random channel approach to disordered kinetics we point out some interesting analogies between colored levy noise and the random channel approach to disordered kinetics these analogies are due to the fact that the probability density of the levy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics although the equations for the two approaches are not identical the analogies can be used for deriving new useful results for both problems the random channel approach makes it possible to generalize the fractional uhlenbeck ornstein processes fuo for space and time dependent colored noise we describe the properties of colored noise in terms of characteristic functionals which are evaluated by using a generalization of huber s approach to complex relaxation phys rev b 31 6070 1985 we start out by investigating the properties of symmetrical white noise and then define the levy colored noise in terms of a langevin equation with a levy white noise source we derive exact analytical expressions for the various characteristic functionals which characterize the noise and a functional fractional fokker planck equation for the probability density functional of the noise at a given moment in time second by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics these equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics

spectral measure of the harmonic oscillator with α stable noise we study the coordinate velocity couple of the harmonic oscillator with stable noise as previously shown by sokolov et al the distribution of this couple is bivariate stable in this work we determine explicitly its associated spectral measure exhibiting directly both the non independence and non ellipticity of the coordinate velocity couple knowledge of the spectral measure allows to analyze and quantify the deviation from ellipticity

continuous and discrete stable processes the one sided levy stable probability densities and the discrete stable distributions form a doubly stochastic poisson transform pair this relationship facilitates the formulation of a class of continuous stable stochastic processes

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

scale affine invariant interest point detectors in this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations our scale and affine invariant detectors are based on the following recent results 1 interest points extracted with the harris detector can be adapted to affine transformations and give repeatable results geometrically stable 2 the characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives the laplacian 3 the affine shape of a point neighborhood is estimated based on the second moment matrix our scale invariant detector computes a multi scale representation for the harris interest point detector and then selects points at which a local measure the laplacian is maximal over scales this provides a set of distinctive points which are invariant to scale rotation and translation as well as robust to illumination changes and limited changes of viewpoint the characteristic scale determines a scale invariant region for each point we extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood an iterative algorithm modifies location scale and neighborhood of each point and converges to affine invariant points this method can deal with significant affine transformations including large scale changes the characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point we present a comparative evaluation of different detectors and show that our approach provides better results than existing methods the performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points

fractional correlation functions in simple viscoelastic liquids abstract we develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time the method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory coupled generalized langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power law memory kernel with long correlation noise the associated fractional fokker planck equation ffpe is also derived and it is shown that it exhibits a fluctuation dissipation theorem fdt the explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive the fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

fluctuations in multiplicative systems with jumps fluctuation properties of the langevin equation including a multiplicative power law noise and a quadratic potential are discussed the noise has the l evy stable distribution if this distribution is truncated the covariance can be derived in the limit of large time it falls exponentially covariance in the stable case studied for the cauchy distribution exhibits a weakly stretched exponential shape and can be approximated by the simple exponential the dependence of that function on system parameters is determined then we consider a dynamics which involves the above process and obeys the generalized langevin equation the same as for the gaussian case the resulting distributions possess power law tails which fall similarly to those for the driving noise whereas central parts can assume the gaussian shape moreover a process with the covariance 1 t at large time is constructed and the corresponding dynamical equation is solved diffusion properties of systems for both covariances are discussed

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

derf distinctive efficient robust features from the biological modeling of the p ganglion cells studies in neuroscience and biological vision have shown that the human retina has strong computational power and its information representation supports vision tasks on both ventral and dorsal pathways in this paper a new local image descriptor termed distinctive efficient robust features derf is derived by modeling the response and distribution properties of the parvocellular projecting ganglion cells in the primate retina derf features exponential scale distribution exponential grid structure and circularly symmetric function difference of gaussian dog used as a convolution kernel all of which are consistent with the characteristics of the ganglion cell array found in neurophysiology anatomy and biophysics in addition a new explanation for local descriptor design is presented from the perspective of wavelet tight frames dog is naturally a wavelet and the structure of the grid points array in our descriptor is closely related to the spatial sampling of wavelets the dog wavelet itself forms a frame and when we modulate the parameters of our descriptor to make the frame tighter the performance of the derf descriptor improves accordingly this is verified by designing a tight frame dog which leads to much better performance extensive experiments conducted in the image matching task on the multiview stereo correspondence data set demonstrate that derf outperforms state of the art methods for both hand crafted and learned descriptors while remaining robust and being much faster to compute

continuous time random walk with correlated waiting times based on the langevin description of the continuous time random walk ctrw we consider a generalization of ctrw in which the waiting times between the subsequent jumps are correlated we discuss the cases of exponential and slowly decaying persistent power law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time in the case of exponential type correlations the sub diffusion at short times is slower than in the absence of correlations at long times the behavior of the mean squared displacement is the same as in uncorrelated ctrw for power law correlations we find subdiffusion characterized by the same exponent at all times which appears to be smaller than the one in uncorrelated ctrw interestingly in the limiting case of an extremely long power law correlations the sub diffusion exponent does not tend to zero but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations

new numerical methods for the riesz space fractional partial differential equations in this paper we consider the numerical solution of the riesz space fractional diffusion equation and advection dispersion equation first a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method furthermore we use the 2 2 pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation and get two difference schemes second using the matrix analysis method we prove that the two difference schemes are unconditionally stable finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

stochastic multi value cellular automata models for bicycle flow in this paper the stochastic randomization is introduced in two different multi value cellular automata ca models in order to model the bicycle flow it is shown that with the randomization effect considered the multiple states in the deterministic multi value ca models disappear and the unique flow density relations fundamental diagrams exist the fundamental diagrams the spacetime plots of the two models are studied in detail it is found that the transition from free flow to congested flow is smooth in one model while it is of second order in the other model the comparison of the results of the two models indicates that in the bicycle flow the priority of the movement should be given to slow bicycles in order to reach a larger maximum flow rate

dasc robust dense descriptor for multi modal and multi spectral correspondence estimation establishing dense correspondences between multiple images is a fundamental task in many applications however finding a reliable correspondence between multi modal or multi spectral images still remains unsolved due to their challenging photometric and geometric variations in this paper we propose a novel dense descriptor called dense adaptive self correlation dasc to estimate dense multi modal and multi spectral correspondences based on an observation that self similarity existing within images is robust to imaging modality variations we define the descriptor with a series of an adaptive self correlation similarity measure between patches sampled by a randomized receptive field pooling in which a sampling pattern is obtained using a discriminative learning the computational redundancy of dense descriptors is dramatically reduced by applying fast edge aware filtering furthermore in order to address geometric variations including scale and rotation we propose a geometry invariant dasc gi dasc descriptor that effectively leverages the dasc through a superpixel based representation for a quantitative evaluation of the gi dasc we build a novel multi modal benchmark as varying photometric and geometric conditions experimental results demonstrate the outstanding performance of the dasc and gi dasc in many cases of dense multi modal and multi spectral correspondences

soluble models for dynamics driven by a super diffusive noise keywords strategie de production first passage time randomly driven harmonic oscillator integrated brownian motion reference lpm article 2002 001view record in web of science record created on 2006 06 22 modified on 2016 08 08

discriminative learning of local image descriptors in this paper we explore methods for learning local image descriptors from training data we describe a set of building blocks for constructing descriptors which can be combined together and jointly optimized so as to minimize the error of a nearest neighbor classifier we consider both linear and nonlinear transforms with dimensionality reduction and make use of discriminant learning techniques such as linear discriminant analysis lda and powell minimization to solve for the parameters using these techniques we obtain descriptors that exceed state of the art performance with low dimensionality in addition to new experiments and recommendations for descriptor learning we are also making available a new and realistic ground truth data set based on multiview stereo data

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

a comparison of affine region detectors the paper gives a snapshot of the state of the art in affine covariant region detectors and compares their performance on a set of test images under varying imaging conditions six types of detectors are included detectors based on affine normalization around harris mikolajczyk and schmid 2002 schaffalitzky and zisserman 2002 and hessian points mikolajczyk and schmid 2002 a detector of maximally stable extremal regions proposed by matas et al 2002 an edge based region detector tuytelaars and van gool 1999 and a detector based on intensity extrema tuytelaars and van gool 2000 and a detector of salient regions proposed by kadir zisserman and brady 2004 the performance is measured against changes in viewpoint scale illumination defocus and image compression the objective of this paper is also to establish a reference test set of images and performance software so that future detectors can be evaluated in the same framework

algebraically decaying noise in a system of particles with hard core interactions we record and analyze the noise experienced by a tracer particle in a one dimensional system of particles interacting with hard core interactions we find that the correlations of the noise are long range with an algebraic decay in time

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

a comparison of dense region detectors for image search and fine grained classification we consider a pipeline for image classification or search based on coding approaches like bag of words or fisher vectors in this context the most common approach is to extract the image patches regularly in a dense manner on several scales this paper proposes and evaluates alternative choices to extract patches densely beyond simple strategies derived from regular interest region detectors we propose approaches based on superpixels edges and a bank of zernike filters used as detectors the different approaches are evaluated on recent image retrieval and fine grained classification benchmarks our results show that the regular dense detector is outperformed by other methods in most situations leading us to improve the state of the art in comparable setups on standard retrieval and fined grained benchmarks as a byproduct of our study we show that existing methods for blob and superpixel extraction achieve high accuracy if the patches are extracted along the edges and not around the detected regions

stable equilibrium based on levy statistics a linear boltzmann equation approach to obtain further insight on possible power law generalizations of boltzmann equilibrium concepts we consider stochastic collision models the models are a generalization of the rayleigh collision model for a heavy one dimensional particle m interacting with ideal gas particles with a mass m m similar to previous approaches we assume elastic uncorrelated and impulsive collisions we let the bath particle velocity distribution function to be of general form namely we do not postulate a specific form of power law equilibrium we show under certain conditions that the velocity distribution function of the heavy particle is levy stable the maxwellian distribution being a special case we demonstrate our results with numerical examples the relation of the power law equilibrium obtained here to thermodynamics is discussed in particular we compare between two models a thermodynamic and an energy scaling approaches these models yield insight into questions like the meaning of temperature for power law equilibrium and into the issue of the universality of the equilibrium i e is the width of the generalized maxwellian distribution functions obtained here independent of coupling constant to the bath

descriptor learning via supervised manifold regularization for multioutput regression multioutput regression has recently shown great ability to solve challenging problems in both computer vision and medical image analysis however due to the huge image variability and ambiguity it is fundamentally challenging to handle the highly complex input target relationship of multioutput regression especially with indiscriminate high dimensional representations in this paper we propose a novel supervised descriptor learning sdl algorithm for multioutput regression which can establish discriminative and compact feature representations to improve the multivariate estimation performance the sdl is formulated as generalized low rank approximations of matrices with a supervised manifold regularization the sdl is able to simultaneously extract discriminative features closely related to multivariate targets and remove irrelevant and redundant information by transforming raw features into a new low dimensional space aligned to targets the achieved discriminative while compact descriptor largely reduces the variability and ambiguity for multioutput regression which enables more accurate and efficient multivariate estimation we conduct extensive evaluation of the proposed sdl on both synthetic data and real world multioutput regression tasks for both computer vision and medical image analysis experimental results have shown that the proposed sdl can achieve high multivariate estimation accuracy on all tasks and largely outperforms the algorithms in the state of the arts our method establishes a novel sdl framework for multioutput regression which can be widely used to boost the performance in different applications

local feature discriminant projection in this paper we propose a novel subspace learning algorithm called local feature discriminant projection lfdp for supervised dimensionality reduction of local features lfdp is able to efficiently seek a subspace to improve the discriminability of local features for classification we make three novel contributions first the proposed lfdp is a general supervised subspace learning algorithm which provides an efficient way for dimensionality reduction of large scale local feature descriptors second we introduce the differential scatter discriminant criterion dsdc to the subspace learning of local feature descriptors which avoids the matrix singularity problem third we propose a generalized orthogonalization method to impose on projections leading to a more compact and less redundant subspace extensive experimental validation on three benchmark datasets including uiuc sports scene 15 and mit indoor demonstrates that the proposed lfdp outperforms other dimensionality reduction methods and achieves state of the art performance for image classification

a note on the fractional hyperbolic differential and difference equations abstract the initial boundary value problem for the fractional differential equation d 2 u t dt 2 d t 1 2 u t au t f t 0 t 1 u 0 0 u 0 in a hilbert space h with the self adjoint positive definite operator a is considered the stability estimates for the solution of this problem and its first and second order derivatives are established the first order of accuracy difference scheme for the approximate solution of this problem is presented the stability estimates for the solution of this difference scheme and its first and second order difference derivatives are established in practice the stability estimates for the solution of difference schemes for one dimensional fractional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional fractional hyperbolic equation with dirichlet condition in space variables are obtained

anomalous fluctuation properties we complement and extend our work on fluctuations arising in nonequilibrium systems in steady states driven by l evy noise h touchette and e g d cohen phys rev e 76 020101 r 2007 as a concrete example we consider a particle subjected to a drag force and a l evy white noise with tail index ensuremath mu 0 2 and calculate the probability distribution of the work done on the particle by the drag force as well as the probability distribution of the work dissipated by the dragged particle in a nonequilibrium steady state for 0l ensuremath mu l2 both distributions satisfy what we call an anomalous fluctuation property characterized by positive and negative fluctuations that asymptotically have the same probability for ensuremath mu 2 by contrast the work and dissipated work distributions satisfy the known conventional and extended fluctuation relations respectively which are both characterized by positive fluctuations that are exponentially more probable than negative fluctuations the difference between these different fluctuation behaviors is discussed in the context of large deviation theory experiments that could probe or reveal an anomalous fluctuation property are also discussed

linearly coupled quantum oscillators with levy stable noise abstract the influence of an external noise on two linearly coupled quantum oscillators is investigated the noise is described by the probability measure corresponding to a symmetric levy stable distribution in calculations the path integral technique is used this paper is a continuation of investigations started by bose datta and feinsilver

primal dual subgradient methods for convex problems in this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure our methods are primal dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem besides other advantages this useful feature provides the methods with a reliable stopping criterion the proposed schemes differ from the classical approaches divergent series methods mirror descent methods by presence of two control sequences the first sequence is responsible for aggregating the support functions in the dual space and the second one establishes a dynamically updated scale between the primal and dual spaces this additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set however we always assume the uniform boundedness of subgradients we present the variants of subgradient schemes for nonsmooth convex minimization minimax problems saddle point problems variational inequalities and stochastic optimization in all situations our methods are proved to be optimal from the view point of worst case black box lower complexity bounds

a new difference scheme with high accuracy and absolute stability for solving convection diffusion equations in this paper we use a semi discrete and a pade approximation method to propose a new difference scheme for solving convection diffusion problems the truncation error of the difference scheme is o h email protected 5 it is shown through analysis that the scheme is unconditionally stable numerical experiments are conducted to test its high accuracy and to compare it with crank nicolson method

multi value cellular automaton models and metastable states in a congested phase in this paper a family of multi value cellular automaton ca associated with traffic flow is presented the family is obtained by extending the rule 184 ca which is an ultradiscrete analogue to the burgers equation ca models in the family show both metastable states and stop and go waves which are often observed in real traffic flow metastable states in the models exist not only on a prominent part of a free phase but also in a congested phase

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

levy noise levy flights levy fluctuations this paper is concerned with a levy noise characterized for t 0 by the functional with 0 2 then levy flights can be defined through a stochastic differential equations rather than the usual chapmann kolmogorov equation we have used this functional approach to solve a plane rotor in the presence of levy noise the linear damped stochastic process driven by levy noise is revisited and its non autonomous and non markovian generalizations have been solved in the context of our functional analysis

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

learning local feature descriptors using convex optimisation the objective of this work is to learn descriptors suitable for the sparse feature detectors used in viewpoint invariant matching we make a number of novel contributions towards this goal first it is shown that learning the pooling regions for the descriptor can be formulated as a convex optimisation problem selecting the regions using sparsity second it is shown that descriptor dimensionality reduction can also be formulated as a convex optimisation problem using mahalanobis matrix nuclear norm regularisation both formulations are based on discriminative large margin learning constraints as the third contribution we evaluate the performance of the compressed descriptors obtained from the learnt real valued descriptors by binarisation finally we propose an extension of our learning formulations to a weakly supervised case which allows us to learn the descriptors from unannotated image collections it is demonstrated that the new learning methods improve over the state of the art in descriptor learning on the annotated local patches data set of brown et al and unannotated photo collections of philbin et al

binary online learned descriptors we propose a novel approach to generate a binary descriptor optimized for each image patch independently the approach is inspired by the linear discriminant embedding that simultaneously increases inter and decreases intra class distances a set of discriminative and uncorrelated binary tests is established from all possible tests in an offline training process the patch adapted descriptors are then efficiently built online from a subset of features which lead to lower intra class distances and thus to a more robust descriptor we perform experiments on three widely used benchmarks and demonstrate improvements in matching performance and illustrate that per patch optimization outperforms global optimization

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction diffusion equations with delay abstract in this paper we study the mild solutions of a class of nonlinear fractional reaction diffusion equations with delay and caputo s fractional derivatives by the measure of noncompactness the theory of resolvent operators the fixed point theorem and the banach contraction mapping principle we develop various theorems for the existence and uniqueness of the global mild solutions for the problem

analytical properties of ultradiscrete burgers equation and rule 184 cellular automaton in this paper we propose an ultradiscrete burgers equation of which all the variables are discrete the equation is derived from a discrete burgers equation under an ultradiscrete limit and reduces to an ultradiscrete diffusion equation through the cole hopf transformation moreover it becomes a cellular automaton ca under appropriate conditions and is identical to rule 184 ca in a specific case we show shock wave solutions and asymptotic behaviours of the ca exactly via the diffusion equation finally we propose a particle model expressed by the ca and discuss a mean flux of particles

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

the exact solution of certain differential equations of fractional order by using operational calculus abstract in the present paper the authors first develop an operational calculus for the familiar riemann liouville fractional differential operator this operational calculus is then used here to solve a cauchy boundary value problem for a certain linear equation involving the riemann liouville fractional derivatives relevant connections are also indicated with the special cases of the equation which were solved earlier by using other methods

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

ldahash improved matching with smaller descriptors sift like local feature descriptors are ubiquitously employed in computer vision applications such as content based retrieval video analysis copy detection object recognition photo tourism and 3d reconstruction feature descriptors can be designed to be invariant to certain classes of photometric and geometric transformations in particular affine and intensity scale transformations however real transformations that an image can undergo can only be approximately modeled in this way and thus most descriptors are only approximately invariant in practice second descriptors are usually high dimensional e g sift is represented as a 128 dimensional vector in large scale retrieval and matching problems this can pose challenges in storing and retrieving descriptor data we map the descriptor vectors into the hamming space in which the hamming metric is used to compare the resulting representations this way we reduce the size of the descriptors by representing them as short binary strings and learn descriptor invariance from examples we show extensive experimental validation demonstrating the advantage of the proposed approach

ornstein uhlenbeck cauchy process we combine earlier investigations of linear systems subject to levy fluctuations with recent attempts to give meaning to so called levy flights in external force fields we give a complete construction of the ornstein uhlenbeck cauchy process as a fully computable paradigm example of doob s stable noise supported ornstein uhlenbeck process despite the nonexistence of all moments we determine local characteristics forward drift of the process generators of forward and backward dynamics and relevant pseudodifferential evolution equations the induced nonstationary spatial process is proved to be markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

a lagrange representation of cellular automaton traffic flow models a new cellular automaton ca model of traffic flow in the lagrange form is proposed in this paper we study the algebraic relationship between models with the lagrange form and the euler form of burger s ca which is constructed from burger s equation using the ultradiscrete method it is found that the lagrange form has made the description of traffic flow in one lane simpler thus we have extended a simple lagrange model to include the effects of inertia of cars and drivers perspective the extended model shows metastable states and complex phase transition from a free to congested state which is similar to the observed data for expressways

full velocity difference model for a car following theory in this paper we present a full velocity difference model for a car following theory based on the previous models in the literature to our knowledge the model is an improvement over the previous ones theoretically because it considers more aspects in car following process than others this point is verified by numerical simulation then we investigate the property of the model using both analytic and numerical methods and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion

on a riesz feller space fractional backward diffusion problem with a nonlinear source abstract in this paper a backward diffusion problem for a space fractional diffusion equation with a nonlinear source in a strip is investigated this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a riesz feller derivative of order 0 2 a nonlinear problem is severely ill posed therefore we propose two new modified regularization solutions to solve it we further show that the approximated problems are well posed and their solutions converge if the original problem has a classical solution in addition the convergence estimates are presented under a priori bounded assumption of the exact solution for estimating the error of the proposed method a numerical example has been implemented