ergodic properties of levy flights coexisting with subdiffusion and related models abstract we analyze ergodic properties of two different stationary processes resulting from a combination of levy flights and subdiffusion the first one comes from the lamperti transformation of subordinated stable process the second one is a sequence of increments of subordinated levy process we prove that both processes are mixing and ergodic we also study increments and the lamperti transformation of time changed fractional brownian motion it appears that these models of anomalous diffusion are also ergodic and mixing we believe that the derived results will help to verify ergodicity in single particle tracking experiments

inverse gaussian and its inverse process as the subordinators of fractional brownian motion in this paper we study the fractional brownian motion fbm time changed by the inverse gaussian ig process and its inverse called the inverse to the inverse gaussian iig process some properties of the time changed processes are similar to those of the classical fbm such as long range dependence however one can also observe different characteristics that are not satisfied by the fbm we study the distributional properties of both subordinators namely ig and iig processes and also that of the fbm time changed by these subordinators we establish also the connections between the subordinated processes considered and the continuous time random walk model for the application part we introduce the simulation procedures for both processes and discuss the estimation schemes for their parameters the effectiveness of these methods is checked for simulated trajectories

age representation of levy walks partial density waves relaxation and first passage time statistics l vy walks lws define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks alternatively there is a hyperbolic representation of them in terms of partial probability density waves using the latter framework we explore the impact of aging on lws which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution we show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear volterra equations for any initial preparation on this basis relaxation properties i e the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics we find that even normal diffusive lws where the long term mean square displacement increases linearly with time may display anomalous relaxation properties such as stretched exponential decay we then discuss the impact of aging on the first passage time statistics of lws by developing the corresponding volterra integral representation as a further natural generalization the concept of lws with wearing is introduced to account for mobility losses

multirate digital signal processing algorithm to calculate complex acoustic pressure fields an efficient algorithm to compute complex magnitude and phase acoustic pressure field data that uses a multirate digital processing architecture is presented the algorithm is based on the discretization of the velocity potential function sampled at a rate that varies as a function of field point location and transducer geometry the algorithm can be used to determine accurate magnitude and phase information at any field point location and for any transducer geometry with a closed form velocity potential function including planar pistons spherically focused pistons and planar annular array transducers e g the nondiffracting or j0 bessel transducer numerical simulations based on this algorithm are presented together with exact field calculations wherever possible in order to make absolute error comparisons additionally results based on a standard gaussian quadrature integration scheme are presented in order to compare computational speed and accuracy in the near field results indicate improv

joint probability distributions for a class of non markovian processes we consider joint probability distributions for the class of coupled langevin equations introduced by fogedby h c fogedby phys rev e 50 1657 1994 we generalize well known results for the single time probability distributions to the case of n time joint probability distributions it is shown that these probability distribution functions can be obtained by an integral transform from distributions of a markovian process the integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non markovian character of the process

forward and backward projection of acoustic fields using fft methods the forward and backward propagation of harmonic acoustic fields using fourier transform methods is presented in particular the forward propagation of a velocity distribution to obtain a pressure field and the backward propagation of a pressure field to obtain a velocity distribution are addressed numerical examples are presented to illustrate the nearfield behavior of the pressure field from complex planar vibrators e g an ultrasonic transducer or plate with nonuniform velocity distributions the numerical results which were obtained via the use of fft algorithms are presented for vibrators which are operating above and below coincidence these results illustrate the acoustic nearfield as a function of distance from the vibrator numerical results are also presented to illustrate the backward projection method the pressure field of a 3 3 focused array is back projected to obtain the velocity distribution for several cases of interest these results illustrate the utility of the transform method and the effect of spatial windows or filters in its implementation using fft algorithms

modifying sqp for degenerate problems most local convergence analyses of the sequential quadratic programming sqp algorithm for nonlinear programming make strong assumptions about the solution namely that the active constraint gradients are linearly independent and that there are no weakly active constraints in this paper we establish a framework for variants of sqp that retain the characteristic superlinear convergence rate even when these assumptions are relaxed proving general convergence results and placing some recently proposed sqp variants in this framework we discuss the reasons for which implementations of sqp often continue to exhibit good local convergence behavior even when the assumptions commonly made in the analysis are violated finally we describe a new algorithm that formalizes and extends standard sqp implementation techniques and we prove convergence results for this method also

fractional levy stable motion can model subdiffusive dynamics we show in this paper that the sample time average mean squared displacement msd of the fractional levy stable motion behaves very differently from the corresponding ensemble average second moment while the ensemble average msd diverges for 2 the sample msd may exhibit either subdiffusion normal diffusion or superdiffusion thus h self similar levy stable processes can model either a subdiffusive diffusive or superdiffusive dynamics in the sense of sample msd we show that the character of the process is controlled by a sign of the memory parameter d h 1 we also introduce a sample p variation dynamics test which allows to distinguish between two models of subdiffusive dynamics finally we illustrate a subdiffusive behavior of the fractional levy stable motion on biological data describing the motion of individual fluorescently labeled mrna molecules inside live e coli cells but it may concern many other fields of contemporary experimental physics doi 10 1103 physreve 82 021130

subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

the van cittert zernike theorem in pulse echo measurements a classical theorem of statistical optics the van cittert zernike theorem is generalized to pulse echo ultrasound this theorem fully describes the second order statistics of the spatial fluctuations the spatial covariance of the field produced by an incoherent source as a random scattering medium is insonified it behaves as an incoherent source the van cittert zernike theorem can thus predict the spatial covariance of the pressure field backscattered by a random medium it is shown that this spatial covariance and the incident energy diagram are fourier pairs in the case of a focused illumination the spatial covariance of the backscattered pressure field is proportional to the autocorrelation of the transmitting aperture function this is independent of frequency and of f number experimental results obtained with a linear array are in good agreement with theoretical expectations the implications of this theorem in speckle reduction and in focusing in nonhomogenous media are discussed

continuous waves generated by focused radiators a single integration technique for calculating the pressure distribution in the field of a focused sinusoidally oscillating radiator is described this technique involves the direct closed form integration of one of the integrals in the rayleigh formula the latter formula itself contains a double integral corresponding to integration over the area of the source transducer face precise experimental measurements of a hydrophone s response to the pressure field generated by a focused medical diagnostic transducer are also reported and comparison between experiment and theory is made very good agreement between theory and experiment is demonstrated with respect to both amplitude and phase for lateral beam profiles at six different distances from the source transducer at a seventh distance about 1 cm from the source transducer much more violent fluctuations in the response are observed than result from the theory

dynamical origin of memory and renewal we show that the dynamic approach to fractional brownian motion fbm establishes a link between a non poisson renewal process with abrupt jumps resetting to zero the system s memory and correlated dynamic processes whose individual trajectories keep a nonvanishing memory of their past time evolution it is well known that the recrossings of the origin by an ordinary one dimensional diffusion trajectory generates a l evy and thus renewal process of index ensuremath theta 1 2 we prove with theoretical and numerical arguments that this is the special case of a more general condition insofar as the recrossings produced by the dynamic fbm generates a l evy process with 0l ensuremath theta l1 this result is extended to produce a satisfactory model for the fluorescent signal of blinking quantum dots

on the sampling conditions for reconstruction of an acoustic field from a finite sound source a simple and accurate method for the estimation of ultrasonic transducer fields is developed in the method the angular spectrum is employed to evaluate the three dimensional propagation from a measured plane to an arbitrary parallel plane the implementation uses a discrete convolution that is described in detail relative to conventional spatial frequency representations the implementation of the angular spectrum method in this paper has the advantage of being free from artifacts enabling sample spacing to be greater than one half wavelength using memory efficiently and interpolating the measured data the loosened sampling requirement and natural interpolation of the method permit efficient reconstruction of the full three dimensional acoustic field from a coarse sound pressure measurement on single plane

analysis of the numerically implemented angular spectrum approach based on the evaluation of two dimensional acoustic fields part ii characteristics as a function of angular range the numerically implemented angular spectrum approach asa is investigated in terms of the characteristics of the asa as a function of angular range over which the angular spectrally decomposed plane waves are superimposed to construct an acoustic field radiated by a source the investigation is based on the evaluation of the 2 d fields radiated by striplike planar sources and 1 d linear phased arrays with different aperture sizes and on the comparison with those obtained by using the analytical solution derived with the point spread function method the convergence and the spatial aliasing of the asa s results are systematically dealt with a region well converging to the analytical solution as the angular range increases is found to exist before the aliasing comes up analytical formulas to determine the boundaries of this region are given based on the instantaneous frequency concept an analytical expression for optimal selection of the angular range used is derived hence the sampling frequency can be chosen to be optimal and adaptive to the discretization size of the source plane without worrying that the number of the samples for the discretization size should be chosen to be a power of two that the fast fourier transform fft usually needs an excellent agreement of the asa s results with the analytical solution is demonstrated when using the optimal selection of the angular range for the specified parameters the analysis made in this paper can be extended to the case of 3 d fields

fractional brownian motion and motion governed by the fractional langevin equation in confined geometries motivated by subdiffusive motion of biomolecules observed in living cells we study the stochastic properties of a non brownian particle whose motion is governed by either fractional brownian motion or the fractional langevin equation and restricted to a finite domain we investigate by analytic calculations and simulations how time averaged observables e g the time averaged mean squared displacement and displacement correlation are affected by spatial confinement and dimensionality in particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain the general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality we define the displacement correlation function and find that this quantity shows distinct features for fractional brownian motion fractional langevin equation and continuous time subdiffusion such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data

superlinear convergence of primal dual interior point algorithms for nonlinear programming the local convergence properties of a class of primal dual interior point methods are analyzed these methods are designed to minimize a nonlinear nonconvex objective function subject to linear equality constraints and general inequalities they involve an inner iteration in which the log barrier merit function is approximately minimized subject to satisfying the linear equality constraints and an outer iteration that specifies both the decrease in the barrier parameter and the level of accuracy for the inner minimization under nondegeneracy assumptions it is shown that asymptotically for each value of the barrier parameter solving a single primal dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements the asymptotic rate of convergence of the resulting algorithm is q superlinear and may be chosen arbitrarily close to quadratic furthermore this rate applies componentwise these results hold in particular for the method described in a r conn n i m gould d orban and p l toint math program ser b 87 2000 pp 215 249 and indicate that the details of its inner minimization are irrelevant in the asymptotics except for its accuracy requirements

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

optimal selection of parameters for the angular spectrum approach to numerically evaluate acoustic fields optimal selection of parameters is presented for the angular spectrum approach asa to the numerical calculation of acoustic fields radiated by planar transducers and linear arrays with and without focusing the parameters include spatial sampling interval discretization size of a source plane in which a source is located and the angular range over which the plane waves decomposed from the source using the discrete fourier transform dft are chosen to superimpose and construct the fields the concept of instantaneous frequency is applied to the fourier transformed green s function to determine the angular range of the plane waves used for the construction with minimal spatial aliasing error based on the minimization of spatial frequency and spatial aliasing errors in a constructed field optimal selection of the parameters is worked out the asa with the optimal selection is then applied to computing the fields radiated by planar transducers and linear arrays into water and a layered oil water medi

anomalous diffusion in correlated continuous time random walks we demonstrate that continuous time random walks in which successive waiting times are correlated by gaussian statistics lead to anomalous diffusion with the mean squared displacement r2 t t2 3 long ranged correlations of the waiting times with a power law exponent 0 2 give rise to subdiffusion of the form r2 t t 1 in contrast correlations in the jump lengths are shown to produce superdiffusion we show that in both cases weak ergodicity breaking occurs our results are in excellent agreement with simulations

when the telegrapher s equation furnishes a better approximation to the transport equation than the diffusion approximation it has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher s equation a j ishimaru appl opt 28 2210 1989 we show that in one dimension the telegrapher s equation furnishes an exact solution to the transport equation in two dimensions we show that since the solution can become negative the telegrapher s equation will not furnish a usable approximation a comparison between simulated data in three dimensions indicates that the solution to the telegrapher s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation s1063 651x 97 04205 0

anomalous diffusion induced by enhancement of memory we introduced simple microscopic non markovian walk models which describe the underlying mechanism of anomalous diffusions in the models we considered the competitions between randomness and memory effects of previous history by introducing the probability parameters the memory effects were considered in two aspects one is the perfect memory of whole history and the other is the latest memory enhanced with time in the perfect memory model superdiffusion was induced with the relation of the hurst exponent h to the controlling parameter p as h p for p 1 2 while in the latest memory enhancement models anomalous diffusions involving both superdiffusion and subdiffusion were induced with the relations h 1 2 and h 1 2 for 0 1 where is the parameter controlling the degree of the latest memory enhancement also we found that although the latest memory was only considered the memory improved with time results in the long range correlations between steps and the correlations increase as time goes on thus we suggest the memory enhancement as a key origin describing anomalous diffusions

evaluation of angle dependent spectral distortion for infinite planar elastic media via angular spectrum decomposition the angle dependent spectral distortion for an infinite planar interface ip asd is examined for the case of fluid fluid and fluid solid elastic interfaces the results are based on computation of the angular spectrum plane wave decomposition of the acoustic pressure field as radiated by an equivalent image source transducer for each frequency of interest the exact incident angle dependent reflection coefficient of the interface may thus be used in the computation versus approximate methods previously required d p orofino and p c pedersen j acoust soc am 92 2883 2899 1992 comparisons of the previous approximate ip asd results to the current exact results are made ip asd results for elastic parallel plates are also presented for a variety of plate orientations and geometries for the special case of normal incidence a comparison of the ip asd result to a previously published result is made the ip asd is additionally examined for a reduced subset of plane wave components that appro

properties of the log barrier function on degenerate nonlinear programs we examine the sequence of local minimizers of the log barrier function for a nonlinear program near a solution at which second order sufficient conditions and the mangasarian fromovitz constraint qualification are satisfied but the active constraint gradients are not necessarily linearly independent when a strict complementarity condition is satisfied we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution and we obtain a semiexplicit characterization of this point when strict complementarity does not hold we obtain several other interesting characterizations in particular an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution

ergodicity breaking and localization we study the joint action of the non poisson renewal events npr yielding continuous time random walk ctrw with index ensuremath alpha l1 and two different generators of hurst coefficient h ensuremath ne 0 5 one generating fractional brownian motion fbm and another scaled brownian motion sbm we discuss the ergodicity breaking emerging from these joint actions and we find that in both cases the adoption of time averages leads to localization in the case of the joint action of npr and sbm localization occurs when sbm would produce subdiffusion the joint action of npr and fbm on the contrary may lead to localization when fbm is a source of superdiffusion the joint action of npr and fbm is equivalent to extending the ctrw to the case where the jumps of the runner are correlated and we argue that the the memory induced localization requires a refinement of the theoretical perspective about determinism and randomness

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

geometric brownian motion with tempered stable waiting times one of the earliest system that was used to asset prices description is black scholes model it is based on geometric brownian motion and was used as a tool for pricing various financial instruments however when it comes to data description geometric brownian motion is not capable to capture many properties of present financial markets one can name here for instance periods of constant values therefore we propose an alternative approach based on subordinated tempered stable geometric brownian motion which is a combination of the popular geometric brownian motion and inverse tempered stable subordinator in this paper we introduce the mentioned process and present its main properties we propose also the estimation procedure and calibrate the analyzed system to real data

some applications of persistent random walks and the telegrapher s equation abstract a persistent random walk can be regarded as a multidimensional markov process the bias free telegraphers equation is 2 p t 2 1 t p t v 2 2 p it can be regarded as interpolating between the wave equation t and the diffusion equation t 0 previously it has found application in thermodynamics cf the review in rev mod phys 61 1989 41 62 1990 375 more recent applications are reviewed in the present article

multipoint correlation functions for continuous time random walk models of anomalous diffusion recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous time random walk ctrw in an external potential two and three point correlation functions are calculated for waiting time distributions with an anomalous power law profile t 1 0 1 on intermediate time scales with a crossover to an exponential long time decay comparison of the ctrw with the brownian harmonic oscillator model gaussian process illustrates how higher order correlation functions may be used to distinguish between dynamical models that have the same two point correlation function

extensions of effective medium theory of transport in disordered systems the effective medium theory of transport in disordered systems whose basis is the replacement of spatial disorder by temporal memory is extended in several practical directions restricting attention to a one dimensional system with bond disorder for specificity a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder it is shown how to use the memory functions in the laplace domain forms in which they first appear and in the time domain forms which are obtained via numerical inversion algorithms to address time evolution of the system beyond the asymptotic domain of large times normally treated an analytic but approximate procedure is provided to obtain the memories in addition to the inversion algorithm good agreement of effective medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit as expected the use of ensemble averages is studied for normal as well as correlation observables the effect of size on effective medium theory is explored and it is shown that even in the asymptotic limit finite size corrections develop to the well known harmonic mean prescription for finding the effective rate a percolation threshold is shown to arise even in one dimension for finite but not infinite systems at a concentration of broken bonds related to the system size spatially long range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories in spite of the fact that the original rates possess nearest neighbor character pausing time distributions in continuous time random walks corresponding to the effective medium memories are calculated

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

memory induced anomalous dynamics emergence of diffusion subdiffusion and superdiffusion from a single random walk model we present a random walk model that exhibits asymptotic subdiffusive diffusive or superdiffusive behavior in different parameter regimes this appears to be an instance of a single random walk model leading to all three forms of behavior by simply changing parameter values furthermore the model offers the great advantage of analytical tractability our model is non markovian in that the next jump of the walker is probabilistically determined by the history of past jumps it also has elements of intermittency in that one possibility at each step is that the walker does not move at all this rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

subdiffusion of mixed origins when ergodicity and nonergodicity coexist single particle trajectories are investigated assuming the coexistence of two subdiffusive processes diffusion on a fractal structure modeling spatial constraints on motion and heavy tailed continuous time random walks representing energetic or chemical traps the particles mean squared displacement is found to depend on the way the mean is taken temporal averaging over single particle trajectories differs from averaging over an ensemble of particles this is shown to stem from subordinating an ergodic anomalous process to a nonergodic one the result is easily generalized to the subordination of any other ergodic process i e fractional brownian motion to a nonergodic one for certain parameters the ergodic diffusion on the underlying fractal structure dominates the transport yet displaying ergodicity breaking and aging

a new class of long tailed pausing time densities for the ctrw we present some asymptotic results for the family of pausing time densities having the asymptotic t property t t ln1 t t 1 in particular we show that for this class of pausing time densities the mean squared displacement r2 t is asymptotically proportional to ln t t and the asymptotic distribution of the displacement has a negative exponential form

nonergodicity of a time series obeying levy statistics time averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied such a process called a levy walk describes dynamical behaviors of many physical systems fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example when the mean sojourn time diverges the process is non ergodic in that case the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function instead it remains random even in the limit of long measurement time several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges and favorably compared to monte carlo simulations nonergodicity of the power spectrum of the process is briefly discussed and a nonstationary wiener khintchine theorem relating the correlation functions and the power spectrum is presented the considered situation is in full contrast to the usual assumptions of ergodicity and stationarity

modeling of nonlinear ultrasound propagation in tissue from array transducers a computationally efficient model capable of simulating finite amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described it is based on a second order operator splitting approach tavakkoli et al j acoust soc am 104 2061 2072 1998 with a fractional step marching scheme whereby the effects of diffraction attenuation and nonlinearity can be computed independently over incremental steps this approach is an extension to that of christopher and parker j acoust soc am 90 507 521 90 488 499 1991 wherein linear and nonlinear effects are propagated separately over incremental steps and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources results of the model are compared with published data predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

localisation and universal fluctuations in ultraslow diffusion processes we study ultraslow diffusion processes with logarithmic mean squared displacement msd comparison of annealed renewal continuous time random walks ctrws with logarithmic waiting time distribution and sinai diffusion in quenched random landscapes reveals striking similarities despite the great differences in their physical nature in particular they exhibit a weakly non ergodic disparity of the time averaged and ensemble averaged msds remarkably for the ctrw we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories we discuss the fundamental connection between the golosov localization effect and non ergodicity in the sense of the disparity between ensemble averaged msd and time averaged msd

numerical evaluation of the rayleigh integral for planar radiators using the fft rayleigh s integral formula is evaluated numerically for planar radiators of any shape with any specified velocity in the source plane using the fast fourier transform algorithm the major advantage of this technique is its speed of computation over 400 times faster than a straightforward two dimensional numerical integration the technique is developed for computation of the radiated pressure in the nearfield of the source and can be easily extended to provide with little computation time the vector intensity in the nearfield computations with the fft of the nearfield pressure of baffled rectangular plates with clamped and free boundaries are compared with the exact solution to illuminate any errors the bias errors introduced by the fft are investigated and a technique is developed to significantly reduce them

weak ergodicity breaking induced by global memory effects we study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states the persistence in a given state or the transition to another one depends on the whole previous temporal history of the system a set of waiting time distributions associated to each state sets the random times between consecutive steps their mean value is finite for all states the probability density of time averaged observables is obtained for different memory mechanisms this statistical object explicitly shows departures between time and ensemble averages while the residence time in each state may have a divergent mean value we demonstrate that this condition is in general not necessary for breaking ergodicity hence we conclude that global memory effects are an alternative mechanism able to induce ergodicity breaking without involving power law statistics analytical and numerical calculations support these results

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

inhomogeneous diffusion and ergodicity breaking induced by global memory effects fil budini adrian adolfo comision nacional de energia atomica gerencia del area de investigaciones y aplicaciones no nucleares gerencia de fisica centro atomico constituyentes argentina consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet patagonia norte argentina

improved estimation of anomalous diffusion exponents in single particle tracking experiments the mean square displacement is a central tool in the analysis of single particle tracking experiments shedding light on various biophysical phenomena frequently parameters are extracted by performing time averages on single particle trajectories followed by ensemble averaging this procedure however suffers from two systematic errors when applied to particles that perform anomalous diffusion the first is significant at short time lags and is induced by measurement errors the second arises from the natural heterogeneity in biophysical systems we show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution as a consequence we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time averaged mean square displacement analysis fails we apply this method to both simulations and in vivo measurements of telomere diffusion in 3t3 mouse embryonic fibroblast cells the motion of telomeres is found to be subdiffusive with an average exponent constant in time individual telomere exponents are normally distributed around the average exponent the proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity

new approaches to the linear propagation of acoustic fields new algorithms are described that provide insight into linear field propagation and offer significant reductions in computational complexity the developments presented here include the usage of a recently developed discrete hankel transform to implement two single step planar propagation algorithms for baffled radially symmetric acoustic pressure or velocity fields an update on the single step approaches that reduce computational complexity through geometrically determined spatial frequency limitations and algorithms for extending to multistep propagation two equivalent means of introducing arbitrary medium attenuation into the above schemes are presented finally a planar boundary crossing algorithm that accounts for refraction and reflection but not multiple reflections is added to one of the multistep propagating algorithms the resulting algorithm is then used to examine the differences between the corresponding fields of a focused piston source operating in water and in a layered fat liver biomedical imaging medium the results yield computationally efficient algorithms that can be used for linear propagation of focused or unfocused beams in attenuating multilayer media and also provide the basis for a novel nonlinear propagation algorithm

extension of the angular spectrum approach to curved radiators the angular spectrum approach asa is conventionally applied to the evaluation of acoustic fields from planar radiators because it is usually based on the 2 d fourier transform or the zero order hankel transform in the axisymmetrical case which is implemented only in a plane the present paper is intended to extend the asa to more general cases where radiators have curved surfaces for this purpose two approaches are developed the first one is the extended asa and is derived in a general way from this approach the angular spectrum of a curved radiator is given by a double integral that does not take the 2 d fourier transform form and thus cannot be implemented using 2 d fast fourier transform fft but by numerical integration the second approach is the indirect asa that gives the angular spectrum via 2 d fourier transforming an initial field pre calculated in a plane the method for calculating the initial field is proposed based on the method developed by ocheltree and frizzell for planar sources an example is given of a linear array with a cylindrically concave surface and in this case the angular spectrum the double integral from the extended asa reduces to a single integral the angular spectra of the array are calculated using both approaches and compared the comparison has shown that the angular spectra obtained from both approaches are in excellent agreement the accuracy and efficiency of the two approaches are studied in the numerical implementation in this example the extended asa has been shown to be more efficient and more accurate than the latter approach both approaches can be applied to arbitrarily curved transducers in the general case where the double integral cannot be reduced to a single integral the latter approach can be more efficient

on the stability of the effective apodization of the nonlinearly generated second harmonic with respect to range the concept of an effective apodization was introduced to describe the field pattern for the nonlinearly generated second harmonic 2f within the focal zone using a linear propagation model our objective in this study was to investigate the validity of the concept of an effective apodization at 2f as an approach to approximating the field of the second harmonic over a wide range of depths two experimental setups were employed a vascular imaging array with a water path and an adult cardiac imaging array with an attenuating liver path in both cases the spatial dependencies of the ultrasonic fields were mapped by scanning a point like hydrophone within a series of planes orthogonal to the propagation direction the sampling distances were located before within and beyond the focal zone the signals were fourier transformed and the complex values at 2f were linearly backpropagated to the transmit plane in order to obtain an effective apodization the measured results demonstrated a relatively constant

overshooting and undershooting subordination scenario for fractional two power law relaxation responses in this paper we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation we investigate properties of a coupled continuous time random walk that follows from modeling the occurrence of jumps with compound counting processes as a result two different diffusion processes corresponding to over and undershooting operational times respectively have been found we show that within the proposed framework all empirical two power law relaxation patterns may be derived this work is motivated by the so called less typical relaxation behavior observed e g for gallium doped cd0 99mn0 01te mixed crystals

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

ergodic properties of fractional brownian langevin motion we investigate the time average mean square displacement overline ensuremath delta 2 mathbf x t mathbf ensuremath int 0 t ensuremath ensuremath delta x t ensuremath ensuremath delta ensuremath x t ensuremath 2 d t ensuremath t ensuremath ensuremath delta for fractional brownian langevin motion where x t is the stochastic trajectory and ensuremath delta is the lag time unlike the previously investigated continuous time random walk model overline ensuremath delta 2 converges to the ensemble average x 2 ensuremath sim t 2h in the long measurement time limit the convergence to ergodic behavior is slow however and surprisingly the hurst exponent h frac 3 4 marks the critical point of the speed of convergence when hl frac 3 4 the ergodicity breaking parameter e b mathbf overline ensuremath delta 2 mathbf x t mathbf mathbf 2 ensuremath overline ensuremath delta 2 mathbf x t mathbf 2 mathbf overline ensuremath delta 2 mathbf x t mathbf 2 ensuremath sim k h ensuremath delta t ensuremath 1 when h frac 3 4 e b ensuremath sim frac 9 16 mathrm ln phantom rule 0 2em 0ex t ensuremath delta t ensuremath 1 and when frac 3 4 lhl1 e b ensuremath sim k h ensuremath delta 4 ensuremath 4h t 4h ensuremath 4 in the ballistic limit h ensuremath rightarrow 1 ergodicity is broken and e b ensuremath sim 2 the critical point h frac 3 4 is marked by the divergence of the coefficient k h fractional brownian motion as a model for recent experiments of subdiffusion of mrna in the cell is briefly discussed and a comparison with the continuous time random walk model is made

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

random walks on bundled structures bundled structures bs are discrete structures obtained joining to each point of a base graph a copy of a fiber graph in condensed matter physics bs are used as realistic models for the geometry and dynamics of nontranslationally invariant systems polymers inhomogeneous systems etc we present an analytical solution for the random walk problem on these structures which is possible when we know the solution for base and fiber separately we obtain an expression for the spectral dimension of the bs as a function of the spectral dimensions of its components moreover we discuss some applications of these results concerning anomalous diffusion laws proving the existence of nondisordered structures with logarithmic and sublogarithmic diffusion laws due only to geometric features

subordinated α stable ornstein uhlenbeck process as a tool for financial data description abstract the classical financial models are based on the standard brownian diffusion type processes however in the exhibition of some real market data like interest or exchange rates we observe characteristic periods of constant values moreover in the case of financial data the assumption of normality is often unsatisfied in such cases the popular vasicek model that is a mathematical system describing the evolution of interest rates based on the ornstein uhlenbeck process seems not to be applicable therefore we propose an alternative approach based on a combination of the popular ornstein uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior the probability density function of the proposed process can be described by a fokker planck type equation and therefore it can be examined as an extension of the basic ornstein uhlenbeck model in this paper we propose the parameters estimation method and calibrate the subordinated vasicek model to the interest rate data

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

fractional telegrapher s equation from fractional persistent random walks we generalize the telegrapher s equation to allow for anomalous transport we derive the space time fractional telegrapher s equation using the formalism of the persistent random walk in continuous time we also obtain the characteristic function of the space time fractional process and study some particular cases and asymptotic approximations similarly to the ordinary telegrapher s equation the time fractional equation also presents distinct behaviors for different time scales specifically transitions between different subdiffusive regimes or from superdiffusion to subdiffusion are shown by the fractional equation as time progresses

elephants can always remember exact long range memory effects in a non markovian random walk we consider a discrete time random walk where the random increment at time step t depends on the full history of the process we calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p at a critical value p 1 c 1 2 where memory effects vanish there is a transition from a weakly localized regime where the walker elephant returns to its starting point to an escape regime inside the escape regime there is a second critical value where the random walk becomes superdiffusive the probability distribution is shown to be governed by a non markovian fokker planck equation with hopping rates that depend both on time and on the starting position of the walk on large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time dependent spring constant k 2p 1 t the solution of this problem is a gaussian distribution with time dependent mean and variance which both depend on the initiation of the process

langevin picture of subdiffusion with infinitely divisible waiting times in this paper we study a langevin approach to modeling of subdiffusion in the presence of time dependent external forces we construct a subordinated langevin process whose probability density function solves the subdiffusive fractional fokker planck equation we generalize the results known for the levy stable waiting times to the case of infinitely divisible waiting time distributions our approach provides a complete mathematical description of subdiffusion with time dependent forces moreover it allows to study the trajectories of the constructed process both analytically and numerically via monte carlo methodology

on fractional tempered stable processes and their governing differential equations we derive the governing equation of the tempered stable subordinator hereafter tss which generalizes the space fractional differential equation satisfied by the law of the stable subordinator itself this equation is expressed in terms of the shifted fractional derivative of order 0 1 coinciding with the stability parameter we then generalize this equation by introducing a time fractional derivative of order 0 1 resp 1 1 and we prove that it is satisfied by the law of a tss time changed by the inverse of a stable subordinator resp by the stable subordinator itself the corresponding processes can therefore be called fractional ts processes finally we provide fractional extensions of the relativistic stable processes which we define as a brownian motion with a random time argument represented by independent fractional ts processes of order resp 1

aging scaled brownian motion scaled brownian motion sbm is widely used to model anomalous diffusion of passive tracers in complex and biological systems it is a highly nonstationary process governed by the langevin equation for brownian motion however with a power law time dependence of the noise strength here we study the aging properties of sbm for both unconfined and confined motion specifically we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak intermediate and strong aging a very rich behavior is revealed for confined aging sbm depending on different aging times and whether the process is sub or superdiffusive we demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale free continuous time random walk processes while sbm exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement finally we derive the density of first passage times in the semi infinite domain that features a crossover defined by the aging time

random walker and the telegrapher s equation a paradigm of a generalized hydrodynamics the telegrapher s equation te is the continuum limit of a persisting random walker we find that the te reproduces the original spectrum almost exactly for all wavelengths far beyond the validity of the expansion this surprising property is used as a paradigm towards the derivation of a generalized hydrodynamics applications to other problems are explored

scaled brownian motion as a mean field model for continuous time random walks we consider scaled brownian motion sbm a random process described by a diffusion equation with explicitly time dependent diffusion coefficient d t d0t 1 batchelor s equation which for 1 is often used for fitting experimental data for subdiffusion of unclear genesis we show that this process is a close relative of subdiffusive continuous time random walks and describes the motion of the rescaled mean position of a cloud of independent walkers it shares with subdiffusive continuous time random walks its nonstationary and nonergodic properties the nonergodicity of sbm does not however go hand in hand with strong difference between its different realizations its heterogeneity ergodicity breaking parameter tends to zero for long trajectories

weak ergodicity breaking in an anomalous diffusion process of mixed origins the ergodicity breaking parameter is a measure for the heterogeneity among different trajectories of one ensemble in this report this parameter is calculated for fractional brownian motion with a random change of time scale often called subordination we show that this quantity is the same as the known continuous time random walks case

infinite densities for levy walks motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion in many cases such systems exhibit strong anomalous diffusion where low order moments ensuremath langle x t q ensuremath rangle with q below a critical value q c exhibit diffusive scaling while for qg q c a ballistic scaling emerges the mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion e g the known diffusion equations and central limit theorems fail to describe both aspects in this paper we resolve this problem by resorting to the concept of infinite density using the widely applicable l evy walk model we find a general expression for the corresponding non normalized density which is fully determined by the particles velocity distribution the anomalous diffusion exponent ensuremath alpha and the diffusion coefficient k ensuremath alpha we explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data

transducer characterization using the angular spectrum method a measurement technique for analyzing the surface velocity patterns of ultrasonic transmitters is presented the technique is based on the angular spectrum method of wave field analysis in this approach acoustic propagation between parallel planar surfaces is modeled using the two dimensional 2 d fourier transform of the wave field with each element in the spatial frequency domain multiplied by the appropriate phase factor the technique was extended from the basic monochromatic model to the wideband pulsed case an experimental system was built to measure the acoustic fields from various transducers including single element and multielement phased arrays backpropagation results are shown for circular planar circular focused and rectangular phase steered transducers the results demonstrate the ability of the extended angular spectrum method to reconstruct the surface velocity distribution of complex acoustic radiators

nonstationary markovian replication process causing diverse diffusions we introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions a nonstationary markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker s displacement and the generalized telegrapher equation with time varying coefficients and we find that diffusivity can be determined by temporal changes of replication of an immediate step by controlling the replications we realize diverse diffusions such as alternating diffusion superdiffusion subdiffusion and marginal diffusion which originate from oscillating increasing decreasing and slowly increasing or decreasing replications with time respectively

the role of linear objective functions in barrier methods we consider the asymptotic behavior of the newton log barrier method for inequality constrained optimization we show that when the objective function is linear an effective step can be taken along the newton direction after each reduction in the barrier parameter leading to efficient performance during the final stages of the algorithm this behavior contrasts with the case of a nonlinear objective where newton s method often performs more and more poorly as the barrier parameter is reduced to zero we analyze the behavior and demonstrate our result on a simple example

formulation for quantitative performance evaluation of holographic imaging in this article an alternative approach is proposed to quantitatively evaluate image quality of holographic reconstructions instead of the conventional approach outlined by the rayleigh resolution criterion this approach describes the image quality in terms of the certainty of the estimation of the source distribution given the classical imaging parameters this formulation gives results in exact agreement with the rayleigh criterion in addition this formulation can be extended to quantitatively evaluate the degradation factors such as aperture offset wave field sampling quantization error noise level and phase only data acquisition that are not considered in the formulation of rayleigh resolution criterion this approach can be also used for quantitative performance evaluation of enhancement algorithms for comparison purposes

analysis of the numerically implemented angular spectrum approach based on the evaluation of two dimensional acoustic fields part i errors due to the discrete fourier transform and discretization the numerically implemented angular spectrum approach asa is investigated based on the evaluation of the 2 d fields radiated by striplike planar sources and 1 d focusing linear phased arrays with different aperture sizes and on the comparison with those obtained by using the analytical solution derived with the point spread function method since the discrete fourier transform dft of a finite size source produces an angular spectrum with spatial frequency aliasing and discretization of the source in even numbered samples causes a half sample length phase shift error in the angular spectrum the effects of the aliasing and the phase shift on calculation accuracy are dealt with the results show that the frequency aliasing causes an overestimation of the near field and the phase shift distorts the results a numerical algorithm for eliminating the aliasing is proposed and correction of the phase shift is confirmed to be necessary the algorithm proposed can completely remove the errors and obtain exact

telegrapher s equations with variable propagation speeds all derivations of the one dimensional telegrapher s equation based on the persistent random walk model assume a constant speed of signal propagation we generalize here the model to allow for a variable propagation speed and study several limiting cases in detail we also show the connections of this model with anomalous diffusion behavior and with inertial dichotomous processes

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

ill conditioning and computational error in interior methods for nonlinear programming ill conditioning has long been regarded as a plague on interior methods but its damaging effects have rarely been documented in fact implementors of interior methods who ignore warnings about the dire consequences of ill conditioning usually manage to compute accurate solutions we offer some insight into this seeming contradiction by analyzing ill conditioning within a primal dual method in which the full usually well conditioned primal dual matrix is transformed to a condensed inherently ill conditioned matrix mpd we show that ill conditioning in the exact condensed matrix closely resembles that known for the primal barrier hessian and then examine the influence of cancellation in the computed constraints using the structure of mpd various bounds are obtained on the absolute accuracy of the computed primal dual steps without cancellation the portion of the computed x step in the small space of mpd a subspace close to the null space of the jacobian of the active constraints has an absolute error bound comparable to machine precision and its large space component has a much smaller error bound with cancellation the usual case the absolute error bounds for both the small and large space components of the computed x step are comparable to machine precision in either case the absolute error bound for the computed multiplier steps associated with active constraints is comparable to machine precision the computed multiplier steps for inactive constraints although converging to zero retain approximately full relative precision because of errors in forming the right hand side the absolute error in the computed solution of the full well conditioned primal dual system is shown to be comparable to machine precision thus under quite general conditions ill conditioning in mpd does not noticeably impair the accuracy of the computed primal dual steps a similar analysis applies to search directions obtained by direct solution of the primal newton equations

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

generalized langevin equation with tempered memory kernel abstract we study a generalized langevin equation for a free particle in presence of a truncated power law and mittag leffler memory kernel it is shown that in presence of truncation the particle from subdiffusive behavior in the short time limit turns to normal diffusion in the long time limit the case of harmonic oscillator is considered as well and the relaxation functions and the normalized displacement correlation function are represented in an exact form by considering external time dependent periodic force we obtain resonant behavior even in case of a free particle due to the influence of the environment on the particle movement additionally the double peak phenomenon in the imaginary part of the complex susceptibility is observed it is obtained that the truncation parameter has a huge influence on the behavior of these quantities and it is shown how the truncation parameter changes the critical frequencies the normalized displacement correlation function for a fractional generalized langevin equation is investigated as well all the results are exact and given in terms of the three parameter mittag leffler function and the prabhakar generalized integral operator which in the kernel contains a three parameter mittag leffler function such kind of truncated langevin equation motion can be of high relevance for the description of lateral diffusion of lipids and proteins in cell membranes

solvable random walk model with memory and its relations with markovian models of anomalous diffusion motivated by studies on the recurrent properties of animal and human mobility we introduce a path dependent random walk model with long range memory for which not only the mean square displacement msd but also the propagator can be obtained exactly in the asymptotic limit the model consists of a random walker on a lattice which at a constant rate stochastically relocates at a site occupied at some earlier time this time in the past is chosen randomly according to a memory kernel whose temporal decay can be varied via an exponent parameter in the weakly non markovian regime memory reduces the diffusion coefficient from the bare value when the mean backward jump in time diverges the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs paradoxically at the transition the process is an anticorrelated l evy flight although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time it belongs to another universality class if memory is very long ranged a second transition takes place to a regime characterized by a logarithmic growth of the msd with time in this case the process is asymptotically gaussian and effectively described as a scaled brownian motion with a diffusion coefficient decaying as 1 t

some properties of the hessian of the logarithmic barrier function more than twenty years ago murray and lootsma showed that hessian matrices of the logarithmic barrier function become increasingly ill conditioned at points on the barrier trajectory as the solution is approached this paper explores some further characteristics of the barrier hessian we first show that except in two special cases the barrier hessian is ill conditioned in an entire region near the solution at points in a more restricted region including the barrier trajectory itself this ill conditioning displays a special structure connected with subspaces defined by the jacobian of the active constraints we then indicate how a cholesky factorization with diagonal pivoting can be used to detect numerical rank deficiency in the barrier hessian and to provide information about the underlying subspaces without making an explicit prediction of the active constraints using this subspace information a close approximation to the newton direction can be calculated by solving linear systems whose condition reflects that of the original problem

on the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy we analyze the convergence of a sequential quadratic programming sqp method for nonlinear programming for the case in which the jacobian of the active constraints is rank deficient at the solution and or strict complementarity does not hold for some or any feasible lagrange multipliers we use a nondifferentiable exact penalty function and we prove that the sequence generated by an sqp using a line search is locally r linearly convergent if the matrix of the quadratic program is positive definite and constant over iterations provided that the mangasarian fromovitz constraint qualification and some second order sufficiency conditions hold