a levinson type algorithm for modeling fast sampled data the standard discrete time autoregressive model is poorly suited for modeling series obtained by sampling continuous time processes at fairly rapid rates large computational errors can occur when the levinson algorithm is used to estimate the parameters of this model because the toeplitz covariance matrix is ill suited for inversion an alternative model is developed based on an incremental difference operator rather than the shift operator it is shown that as the sampling period goes to zero unlike the standard autoregressive parameters the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous time process a levinson type algorithm for efficiently estimating the parameters of this model is derived numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the levinson algorithm

conformable derivative approach to anomalous diffusion abstract by using a new derivative with fractional order referred to conformable derivative an alternative representation of the diffusion equation is proposed to improve the modeling of anomalous diffusion the analytical solutions of the conformable derivative model in terms of gauss kernel and error function are presented the power law of the mean square displacement for the conformable diffusion model is studied invoking the time dependent gauss kernel the parameters related to the conformable derivative model are determined by levenberg marquardt method on the basis of the experimental data of chloride ions transportation in reinforced concrete the data fitting results showed that the conformable derivative model agrees better with the experimental data than the normal diffusion equation furthermore the potential application of the proposed conformable derivative model of water flow in low permeability media is discussed

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

estimation of continuous time stochastic signals from sample covariances the problem of estimating the parameters in stochastic continuous time signals represented as continuous time autoregressive moving average arma processes from discrete time data is considered the proposed solution is to fit the covariance function of the process parameterized by the unknown parameters to sample covariances it is shown that the method is consistent and an expression for the approximate covariance matrix of the estimated parameter vector is derived the derived variances are compared with empirical variances from a monte carlo simulation and with the cramer rao bound it turns out that the variances are close to the cramer rao bound for certain choices of the sampling interval and the number of covariance elements used in the criterion function

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on the stability of a general polynomial in the space of markov parameters abstract markov parameters and the associated stability criterion were first introduced for continuous time real polynomials recently robust stability of such polynomials was considered in markov parameters space where efficient robust stability tests were obtained based on the markov theorem this has motivated the authors to extend the above idea to more general types of polynomials and develop markov parameters and the associated stability criterion for complex continuous time as well as real and complex discrete time polynomials moreover for each polynomial type we present compact relations in order to recover the coefficients of a polynomial corresponding to a given set of markov parameters the stability results presented here may be useful for 1 d and 2 d filtering and control applications

asymptotic green s functions for time fractional diffusion equation and their application for anomalous diffusion problem abstract asymptotic green s functions for short and long times for time fractional diffusion equation derived by simple and heuristic method are provided in case if fractional derivative is presented in caputo sense the applicability of the asymptotic green s functions for solving the anomalous diffusion problem on a semi infinite rod is demonstrated the initial value problem for longtime solution of the time fractional diffusion equation by green s function approach is resolved

an efficient order recursive algorithm with a lattice structure for estimating continuous time ar process parameters abstract an order recursive algorithm is proposed to solve the normal equation in estimating continuous time ar process parameters the new algorithm requires o m 2 computations where m is the order of the ar process and therefore is computationally efficient this is similar to the levinson type algorithm of pham and le breton however the proposed algorithm has a new associated lattice structure and as a consequence a new stability test for continuous time systems not just ar processes by checking the reflection coefficients of the lattice this can be viewed as a continuous time analogy that is recursive in order of the well known discrete time version its discrete time approximation when sampled data are given and connections with other algorithms are also discussed

a delta mywe algorithm for parameter estimation of noisy ar processes we develop a delta operator based modified yule walker equation algorithm mywe for parameter estimation of a noisy autoregressive ar process the methodology in developing this new algorithm is similar to the previous works on pure ar processes computer simulation results are given to show the improvement of performance in estimating ar parameters in white noise over the q operator mywe algorithm

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

efficient solution of linear systems of equations with recursive structure abstract inversion algorithms of order n2 for n n matrices with recursive structure are derived in a unified way several known algorithms as well as some new ones for toeplitz type hilbert type and vandermonde type matrices are given

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

rational solutions for the time fractional diffusion equation a uniform rational approximation of the mittag leffler function is derived which serves as a global approximant accounting for both the taylor series for small arguments and asymptotic series for

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

fractional diffusion equation in a confined region surface effects and exact solutions surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated first we consider the one dimensional case with the boundary conditions rho 0 t phi0 t and rho a t phia t subsequently the two dimensional case in the cylindrical symmetry with rho a theta t phia theta t and rho b theta t phib theta t is investigated for these cases we also obtain exact solutions for an arbitrary initial condition by using the green s function approach

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems