a recency effect in sound localization in a free field pointing task listeners localized trains of 4 32 spatially distributed gabor clicks narrowband impulses centered at 4 khz carrier frequency and repeating at an interval of 5ms multiple regression coefficients estimated the perceptual weight applied to each click in a train during location judgments temporal weighting functions obtained in this way exhibited two key features onset dominance as evidenced by high weight on the initial click and upweighting of late arriving sound as evidenced by weights that gradually increased over the duration of each click train across all tested click train durations and despite randomly varying the durations from trial to trial the greatest post onset weights were consistently found for clicks at or near the offset the results imply a special importance of late arriving sound rather than feedforward recovery from onset dominance and are broadly consistent with recency effects resulting from temporal integration

fast finite difference approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients in this paper we consider an inverse problem for identifying the fractional derivative indices in a two dimensional space fractional nonlocal model based on a generalization of the two sided riemann liouville formulation with variable diffusivity coefficients first we derive an implicit difference method idm for the direct problem and the stability and convergence of the idm are discussed second for the implementation of the idm we develop a fast bi conjugate gradient stabilized method fbi cgstab that is superior in computational performance to gaussian elimination and attains the same accuracy third we utilize the levenberg marquardt l m regularization technique combined with the armijo rule the popular inexact line search condition to solve the modified nonlinear least squares model associated with the parameter identification finally we carry out numerical tests to verify the accuracy and efficiency of the idm numerical investigations are performed with both accurate data and noisy

numerical approximation of a one dimensional space fractional advection dispersion equation with boundary layer finite element computations for singularly perturbed convection diffusion equations have long been an attractive theme for numerical analysis in this article we consider the singularly perturbed fractional advection dispersion equation fade with boundary layer behavior we derive a theoretical estimate which shows that the under resolved case corresponds to e

a geometric deletion restriction formula abstract in this paper we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement the logarithmic ideal was introduced in cohen et al 2011 5 in a study of the critical points of the master function the above result is used to understand the asymptotic behaviour of the hilbert series of the logarithmic ideal as an application we note that a well known formula due to solomon and terao may be expressed as an euler characteristic and at least in the case of tame arrangements deduced from our main theorem

intersection numbers of twisted cycles and the correlation functions of the conformal field theory the coefficients of the four point correlation functions calculated by dotsenko fateev are interpreted naturally in terms of the intersection numbers of twisted or loaded cycles this is an answer to the long standing problem of clarifying the meaning of such coefficients appearing in correlation functions

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

a fourth order compact adi scheme for two dimensional nonlinear space fractional schrodinger equation in this paper a novel compact operator is derived for the approximation of the riesz derivative with order alpha in 1 2 the compact operator is proved with fourth order accuracy combining the compact operator in space discretization a linearized difference scheme is proposed for a two dimensional nonlinear space fractional schrodinger equation it is proved that the difference scheme is uniquely solvable stable and convergent with order o tau 2 h 4 where tau is the time step size h max h 1 h 2 and h 1 h 2 are space grid sizes in the x direction and the y direction respectively based on the linearized difference scheme a compact alternating direction implicit scheme is presented and analyzed numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

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

temporal weighting of interaural time and level differences in high rate click trains temporal weighting functions twfs quantifying sensitivity to interaural time differences itd and interaural level differences ild over the duration of brief stimuli were measured in 6 normal hearing subjects using trains of 16 gabor clicks centered at 4 khz presented dichotically at 4 rates inter click intervals ici of 10 5 2 5 and 1 25 ms random itd or ild were imposed independently on each click in the train in separate conditions the subject s task was to discriminate the lateral position of the click train left or right receiver operating characteristic roc analysis was then used to quantify the effectiveness or weight of each click according to individual click itd or ild although individual differences were evident onset cues appeared to dominate at high rates onset dominance was apparent for both itd and ild at 1 25 ms ici and for itd at 2 5 ms ici but for neither cue at 5 or 10 ms ici onset dominance was greater on average for itd than ild although twfs were qualitatively similar for the two cues no evidence was found for upweighting of late arriving ild stecker g c and hafter e r 2009 j acoust soc am 125 3914 3924

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

temporal effects in interaural and sequential level difference perception temporal effects in interaural level difference ild perception are not well understood while it is often assumed that ild sensitivity is independent of the temporal stimulus properties a reduction of ild sensitivity for stimuli with a high modulation rate has been reported known under the term binaural adaptation experiment 1 compared ild thresholds and sequential level difference sld thresholds using 300 ms bandpass filtered pulse trains centered at 4 khz with rates of 100 400 and 800 pulses per second pps in contrast to the sld thresholds ild thresholds were elevated at 800 pps consistent with literature data that had previously been attributed to binaural adaptation experiment 2 showed better ild sensitivity for pulse trains than for pure tones suggesting that amplitude modulation enhances ild sensitivity the present ild data and binaural adaptation data from the literature were predicted by a model combining well established auditory periphery front ends with an interaural comparison stage the model also accounted for other published ild data including target ild thresholds in diotic forward and backward fringes and ild thresholds with different amounts of interaural correlation overall a variety of temporal effects in ild perception including binaural adaptation appear to be largely attributable to monaural peripheral auditory processing

galerkin spectral method for nonlinear time fractional cable equation with smooth and nonsmooth solutions abstract in this work we study the numerical solutions of the time fractional cable equations with nonlinear term where the fractional derivatives are described in riemann liouville sense an explicit scheme is constructed based upon finite difference method in time and legendre spectral method in space stability and convergence of scheme are proved rigorously moreover an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first order derivative fractional derivatives and nonlinear term numerical examples are given to support theoretical analysis

implicit explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions we propose second order implicit explicit imex time stepping schemes for nonlinear fractional differential equations with fractional order 0 beta 1 from the known structure of the nonsmooth solution and by introducing corresponding correction terms we can obtain uniformly second order accuracy from these schemes we prove the convergence and linear stability of the proposed schemes numerical examples illustrate the flexibility and efficiency of the imex schemes and show that they are effective for nonlinear and multirate fractional differential systems as well as multiterm fractional differential systems with nonsmooth solutions

convergence of a p version hp version method for fractional differential equations recently m zayernouri and g e karniadakis 2014 10 proposed a new spectral method for fractional differential equations and observed an exponential rate of convergence in this paper we will prove a convergence rate of their spectral method and thus provide an explanation for their observations

the role of envelope statistics in detecting changes in interaural correlation the role of envelope statistics in binaural masking level differences bmlds and correlation change detection was investigated in normal hearing listeners thresholds and just noticeable differences jnds were measured for different bandwidths and center frequencies 500 2000 4000 and 8000 hz using gaussian noises gns and low fluctuation noises lfns at a 500 hz center frequency gn noso thresholds were higher than nos thresholds were lower than and correlation change detection jnds were the same as lfn thresholds and jnds at higher center frequencies gn nos thresholds were the same or higher than lfn thresholds and gn correlation change detection jnds were much smaller than lfn jnds using a pulsed sine vocoder a second experiment was performed to investigate if binaural adaptation might contribute to the difference in gn and lfn detection there was no effect of pulse rate thus providing no clear evidence that binaural adaptation plays a role in these tasks both a cross correlation model and a model that utilized the fluctuations in the interaural differences could explain a majority of the variance in the lfn correlation change jnds

intersection numbers of twisted cycles associated with the selberg integral and an application to the conformal field theory intersection numbers of twisted or loaded cycles associated with the selberg integral are studied in particular the self intersection number of the cycle which is invariant under the action of the symmetric group is expressed by the product of trigonometric functions this formula reproduces the four point correlation functions in the conformal field theory calculated by dotsenko fateev in 3 in our study a compact non singular model terada model of the configuration space of n 3 points on the real projective line and a q analogue of the chu vadermonde formula for the hypergeometric series play a crucial role intersection numbers of the corresponding cocycles are also studied

detection of interaural differences of intensity in trains of high frequency clicks as a function of interclick interval and number listeners were asked to detect interaural differences of time in trains of 4000 hz clicks as the interclick interval ici was varied from 10 to 1 ms and the number of clicks in a train n was varied from 1 to 32 plots of log interaural threshold versus log n produce straight lines whose absolute slopes decrease toward 0 0 with decreasing ici these results are shown to fit a saturation model which argues that as the click rate increases the evoked neural activity moves from a response that is tonic toward one which is more phasic the need to postulate neural compression is based in part on the fact that the three most commonly cited models of the limitations imposed by high frequency reduction in the depth of modulations due to narrow band filtering within the auditory system neural refractoriness and nonindependence of successive samples of internal noise do not predict a change in slope with rate

second order numerical methods for multi term fractional differential equations smooth and non smooth solutions abstract starting with the asymptotic expansion of the error equation of the shifted grunwald letnikov formula we derive a new modified weighted shifted grunwald letnikov wsgl formula by introducing appropriate correction terms we then apply one special case of the modified wsgl formula to solve multi term fractional ordinary and partial differential equations and we prove the linear stability and second order convergence for both smooth and non smooth solutions when the regularity of the solutions is known we show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms moreover the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions numerical simulations verify the theoretical results and demonstrate that the new wsgl formula leads to better performance compared to other known numerical approximations with similar resolution

a fast semi implicit difference method for a nonlinear two sided space fractional diffusion equation with variable diffusivity coefficients in this paper we derive a new nonlinear two sided space fractional diffusion equation with variable coefficients from the fractional fick s law a semi implicit difference method sidm for this equation is proposed the stability and convergence of the sidm are discussed for the implementation we develop a fast accurate iterative method for the sidm by decomposing the dense coefficient matrix into a combination of toeplitz like matrices this fast iterative method significantly reduces the storage requirement of o n 2 and computational cost of o n 3 down to n and o n log n where n is the number of grid points the method retains the same accuracy as the underlying sidm solved with gaussian elimination finally some numerical results are shown to verify the accuracy and efficiency of the new method

novel numerical methods for solving the time space fractional diffusion equation in two dimensions in this paper a time space fractional diffusion equation in two dimensions tsfde 2d with homogeneous dirichlet boundary conditions is considered the tsfde 2d is obtained from the standard diffusion equation by replacing the first order time derivative with the caputo fractional derivative t d gamma gamma in 0 1 and the second order space derivatives with the fractional laplacian delta alpha 2 alpha in 1 2 using the matrix transfer technique proposed by ilic et al m ilic f liu i turner and v anh fract calc appl anal 9 2006 pp 333 349 the tsfde 2d is transformed into a time fractional differential system as t d gamma mathbf u k alpha mathbf a alpha 2 mathbf u where mathbf a is the approximate matrix representation of delta traditional approximation of mathbf a alpha 2 requires diagonalization of mathbf a which is very time consuming for large sparse matrices the novelty of our proposed numerical schemes is that using either the finite difference method or the laplace transform to handle the caputo time fractional derivative the solution of the tsfde 2d is written in terms of a matrix function vector product f mathbf a mathbf b at each time step where mathbf b is a suitably defined vector depending on the method used to generate the matrix mathbf a the product f mathbf a mathbf b can be approximated using either the preconditioned lanczos method when mathbf a is symmetric or the mathbf m lanzcos method when mathbf a is nonsymmetric which are powerful techniques for solving large linear systems we give error bounds for the new methods and illustrate their roles in solving the tsfde 2d we also derive the analytical solution of the tsfde 2d in terms of the mittag leffler function finally numerical results are presented to verify the proposed numerical solution strategies

a simple finite element method for boundary value problems with a riemann liouville derivative we consider a boundary value problem involving a riemann liouville fractional derivative of order 3 2 2 on the unit interval 0 1 the standard galerkin finite element approximation converges slowly due to the presence of singularity term x 1 in the solution representation in this work we develop a simple technique by transforming it into a second order two point boundary value problem with nonlocal low order terms whose solution can reconstruct directly the solution to the original problem the stability of the variational formulation and the optimal regularity pickup of the solution are analyzed a novel galerkin finite element method with piecewise linear or quadratic finite elements is developed and l 2 d error estimates are provided the approach is then applied to the corresponding fractional sturm liouville problem and error estimates of the eigenvalue approximations are given extensive numerical results fully confirm our theoretical study

stability and convergence of a finite volume method for the space fractional advection dispersion equation we consider the space fractional advection dispersion equation which is obtained from the classical advection diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order we derive a finite volume method that utilises fractionally shifted grunwald formulae for the discretisation of the fractional derivative to numerically solve the equation on a finite domain with homogeneous dirichlet boundary conditions we prove that the method is stable and convergent when coupled with an implicit timestepping strategy results of numerical experiments are presented that support the theoretical analysis

finite difference methods for the time fractional diffusion equation on non uniform meshes since fractional derivatives are integrals with weakly singular kernel the discretization on the uniform mesh may lead to poor accuracy the finite difference approximation of caputo derivative on non uniform meshes is investigated in this paper the method is applied to solve the fractional diffusion equation and a semi discrete scheme is obtained the unconditional stability and h 1 norm convergence are proved a fully discrete difference scheme is constructed with space discretization by compact difference method the error estimates are established for two kinds of nonuniform meshes numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods moreover a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

uniform estimates for eulerian lagrangian methods for singularly perturbed time dependent problems we prove a priori optimal order error estimates in a weighted energy norm for several eulerian lagrangian methods for singularly perturbed time dependent convection diffusion equations with full regularity the estimates depend only on certain sobolev norms of the initial and right hand side data but not on varepsilon or any norm of the true solution and so hold uniformly with respect to varepsilon we use the interpolation of spaces and stability estimates to derive an varepsilon uniform estimate for problems with minimal or intermediate regularity where the convergence rates are proportional to certain besov norms of the initial and right hand side data

onset and offset specific effects in interaural level difference discrimination the relative sensitivity of human listeners to interaural level differences ilds carried by the onsets offsets and interior portions of brief sounds was examined stimuli consisted of single 4000 hz gabor clicks gaussian windowed tone bursts or trains of 16 such clicks repeating at an interclick interval ici of 2 or 5 ms in separate conditions ilds favored the right ear by a constant amount for all clicks condition rrrr or a changing amount that was maximal at sound onset condition r000 offset condition 000r both onset and offset condition r00r or at the temporal midpoint of the stimulus condition 0rr0 ild increases and decreases were implemented as linear decibel sweeps across four clicks to minimize transient distortion threshold ilds were determined adaptively for each of these conditions and for single clicks thresholds were similar for ilds presented near sound onset or offset condition r000 vs 000r but lower when ilds were carried by both onset and offset clicks condition r00r than for ilds carried by interior clicks alone condition 0rr0 the results suggest that similar sensitivity to onset and offset ild does not reflect uniform temporal weighting instead ild sensitivity favors onsets and offsets over the interior portions of sounds

a preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh numerical methods for fractional differential equations generate full stiffness matrices which were traditionally solved via gaussian type direct solvers that require o n 3 of computational work and o n 2 of memory to store where n is the number of spatial grid points in the discretization we develop a preconditioned fast krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions numerical results are presented to show the utility of the method

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

a novel compact adi scheme for two dimensional riesz space fractional nonlinear reaction diffusion equations abstract this paper is concerned with the construction and analysis of a novel linearized compact adi scheme for the two dimensional riesz space fractional nonlinear reaction diffusion equations convergence of the proposed scheme is proved the highlight is that the time discretization is achieved by applying a second order one step and linearized method the time discretization requires only one starting value which is sharp contrast to the extrapolated crank nicolson method or the usual second order linearized schemes numerical examples on several fractional models are presented to confirm our theoretical results

chern classes for singular hypersurfaces we prove a formula expressing the chern schwartz macpherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology chern class of singular varieties introduced by w fulton and we discuss the relation between these classes and others such as mather s chern class and the l class we introduced in previous work 0 introduction there are several candidates for a notion of homology chern class of a possibly singular algebraic variety x agreeing with the class obtained as dual of the total chern class of the tangent bundle of x if x is nonsingular one such notion was introduced by r macpherson macpherson agreeing as it was later understood with a class introduced earlier by m ij schwartz and enjoying good functorial properties a different class was defined by w fulton fulton 4 2 6 in a different and somewhat more general setting the main purpose of this paper is to prove a precise relation between the schwartz macpherson and the fulton class of a hypersurface x of a nonsingular variety we denote these two classes by csm x cf x respectively a summary of the context and essential definitions is given in 1 where we present several different statements of the result the version which expresses most directly the link between the two classes mentioned above goes as follows fulton s class for a subscheme x of a nonsingular variety m is defined by cf x m c tmix n s x m that is by capping the total chern class of the ambient space against the segre class of the subscheme it is proved in fulton 4 2 6 that this definition does not depend on the choice of the ambient variety m for a hypersurface x of a nonsingular variety m let y be the singular subscheme of x locally defined by the partial derivatives of an equation of x for an integer k 0 we can consider the k th thickening x k of x along y and then consider its fulton class cf x k it is easily seen that cf x k is a polynomial in k with coefficients in the chow group a x so it can be formally evaluated at negative k s we can then define a class c x cf x 1 the main result of this paper is theorem c x is equal to the chern schwartz macpherson class of x received by the editors june 3 1997 1991 mathematics subject classification prinmary 14c17 32s60 supported in part by nsf grant dms 9500843 1999 american mathbemiatical society 3989 this content downloaded from 157 55 39 136 on thu 01 dec 2016 05 45 16 utc all use subject to http about jstor org terms

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

a petrov galerkin finite element method for fractional convection diffusion equations in this work we develop variational formulations of petrov galerkin type for one dimensional fractional boundary value problems involving either a riemann liouville or caputo derivative of order alpha in 3 2 2 in the leading term and both convection and potential terms they arise in the mathematical modeling of asymmetric superdiffusion processes in heterogeneous media the well posedness of the formulations and sharp regularity pickup of the variational solutions are established a novel finite element method fem is developed which employs continuous piecewise linear finite elements and shifted fractional powers for the trial and test space respectively the new approach has a number of distinct features it allows the derivation of optimal error estimates in both the l 2 d and h 1 d norms and on a uniform mesh the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned further in the riemann liouville case an enriched fem is propose

spectro temporal weighting of interaural time differences in speech numerous studies have demonstrated that the perceptual weighting of interaural time differences itds is non uniform in time and frequency leading to reports of spectral and temporal dominance regions it is unclear however how these dominance regions apply to spectro temporally complex stimuli such as speech the authors report spectro temporal weighting functions for itds in a pair of naturally spoken speech tokens two and eight each speech token was composed of two phonemes and was partitioned into eight frequency regions over two time bins one time bin for each phoneme to derive lateralization weights itds for each time frequency bin were drawn independently from a normal distribution with a mean of 0 and a standard deviation of 200 s and listeners were asked to indicate whether the speech token was presented from the left or right itd thresholds were also obtained for each of the 16 time frequency bins in isolation the results suggest that spectral dominance regions apply to speech and that itds carried by phonemes in the first position of the syllable contribute more strongly to lateralization judgments than itds carried by phonemes in the second position the results also show that lateralization judgments are partially accounted for by itd sensitivity across time frequency bins

finite difference methods with non uniform meshes for nonlinear fractional differential equations in this article finite difference methods with non uniform meshes for solving nonlinear fractional differential equations are presented where the non equidistant stepsize is non decreasing the rectangle formula and trapezoid formula are proposed based on the non uniform meshes combining the above two methods we then establish the predictor corrector scheme the error and stability analysis are carefully investigated at last numerical examples are carried out to verify the theoretical analysis besides the comparisons between non uniform and uniform meshes are given where the non uniform meshes show the better performance when dealing with the less smooth problems

vector space of feynman integrals and multivariate intersection numbers feynman integrals obey linear relations governed by intersection numbers which act as scalar products between vector spaces we present a general algorithm for the construction of multivariate intersection numbers relevant to feynman integrals and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections and to directly derive functional equations fulfilled by the latter we apply it to the decomposition of a few feynman integrals at one and two loops as first steps toward potential applications to generic multiloop integrals the proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations

a generalized spectral collocation method with tunable accuracy for fractional differential equations with end point singularities we develop spectral collocation methods for fractional differential equations with variable order with two end point singularities specifically we derive three term recurrence relations for both integrals and derivatives of the weighted jacobi polynomials of the form 1 x mu 1 1 x mu 2 p j a b x a b mu 1 mu 2 1 which leads to the desired differentiation matrices we apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end point singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters mu 1 and mu 2 even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method

numerical algorithms for time fractional subdiffusion equation with second order accuracy this article aims to fill in the gap of the second order accurate schemes for the time fractional subdiffusion equation with unconditional stability two fully discrete schemes are first proposed f

stability and convergence of finite difference schemes for a class of time fractional sub diffusion equations based on certain superconvergence this paper is devoted to the construction and analysis of finite difference methods for solving a class of time fractional subdiffusion equations based on the certain superconvergence at some particular points of the fractional derivative by the traditional first order grunwald letnikov formula some effective finite difference schemes are derived the obtained schemes can achieve the global second order numerical accuracy in time which is independent of the values of anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme respectively are established for the one dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method furthermore the extension to the two dimensional case is also considered numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes the superconvergent points of the traditional first order grunwald letnikov formula to approximate the riemann liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time fractional sub diffusion equations the obtained schemes can achieve the global second order accuracy in time which is independent of the anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme are presented respectively the strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method the extension to the two dimensional case is also discussed the compact difference scheme with the convergence of second order in time and fourth order in space is derived along with the strict stability and convergence analysis several numerical experiments are implemented to illustrate the computational efficiency numerical accuracy and robustness of the proposed schemes

temporal weighting functions for interaural time and level differences iii temporal weighting for lateral position judgments temporal variation in listeners sensitivity to interaural time and level differences itd and ild was assessed using the temporal weighting function twf paradigm stecker and hafter 2002 j acoust soc am 112 1046 1057 in the context of sound source lateralization brief gabor click trains were presented over headphones with overall itd and or ild ranging 500 s itd and or 5 db ild across trials values for individual clicks within each train varied by an additional 100 s or 2 db to allow twf calculation by multiple regression in separate conditions twfs were measured for i itd alone ii ild alone iii itd and ild covarying in agreement and iv itd and ild varying independently across clicks consistent with past studies that measured twf for binaural discrimination twfs demonstrated high weight on the first click for stimuli with short interclick interval ici 2 ms but flatter weighting for longer ici 5 10 ms some conditions additionally demonstrated greater weight for clicks near the offset than near the middle of the train stecker and hafter 2009 j acoust soc am 125 3914 3924 the latter result was observed only when stimuli carried ild and appeared more reliably for 5 ms than for 2 or 10 ms ici

detection and discrimination of simulated motion of auditory targets in the horizontal plane three experiments investigated subjects ability to detect and discriminate the simulated horizontal motion of auditory targets in an anechoic environment moving stimuli were produced by dynamic application of stereophonic balancing algorithms to a two loudspeaker system with a 30 separation all stimuli were 500 hz tones in experiment 1 subjects had to discriminate a left to right moving stimulus from a stationary stimulus pulsed for the same duration 300 or 600 ms for both durations minimum audible movement angles mama s were on the order of 5 for stimuli presented at 0 azimuth straight ahead and increased to greater than 30 for stimuli presented at 90 azimuth experiment 2 further investigated mama s at 0 azimuth employing two different procedures to track threshold holding stimulus duration constant at 100 600 ms while varying velocity or holding the velocity constant at 22 360 s while varying duration results from the two procedures agreed with each other and

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

temporal weighting of binaural cues revealed by detection of dynamic interaural differences in high rate gabor click trains listeners detected interaural differences of time itds or level ilds carried by single 4000 hz gabor clicks gaussian windowed tone bursts and trains of 16 such clicks repeating at an interclick interval ici of 2 5 or 10 ms in separate conditions target interaural differences favored the right ear by a constant amount for all clicks condition rr attained their peak value at onset and diminished linearly to 0 at offset condition r0 or grew linearly from 0 at onset to a peak value at offset condition 0r threshold itds and ilds were determined adaptively in separate experiments for each of these conditions and for single clicks itd thresholds were found to be lower for 16 click trains than for single clicks at 10 ms ici regardless of stimulus condition at 2 ms ici thresholds in rr and r0 conditions were similar to single click thresholds at 2 ms ici thresholds in the 0r condition were significantly worse than for single clicks at 2 ms ici consistent with strong rate dependent onset dominance in listeners temporal weighting of itd ild thresholds in contrast were predominantly unaffected by ici suggesting little or no onset dominance for ild of high rate stimuli

optimal convergence rates for semidiscrete finite element approximations of linear space fractional partial differential equations under minimal regularity assumptions abstract we consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by jin et al 2015 and ervin et al 2018 the error estimates are proved by using two approaches one approach is to apply the duality argument in johnson 1987 for the heat equation to consider the error estimates for the linear space fractional partial differential equations this argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution another approach is to use the approximate solution operators of the corresponding fractional elliptic problems this argument can be extended to consider more general linear space fractional partial differential equations numerical examples are given to show that the numerical results are consistent with the theoretical results

a tunable finite difference method for fractional differential equations with non smooth solutions in this work a finite difference method of tunable accuracy for fractional differential equations fdes with end point singularities is developed modified weighted shifted grunwald letnikov wsgl formulas are proposed to approximate the left and right riemann liouville fractional operators which show better accuracy than the original wsgl formulas due to the use of the correction terms finite difference schemes are constructed to solve two fractional boundary value problems and a space fractional allen cahn equation even if the singularity of the considered fdes is unknown satisfactory numerical solutions can still be obtained by suitably tuning the correction terms various numerical examples are presented to verify the effectiveness of the present method and comparisons with other known methods are also made that demonstrate higher accuracy of the current method

stability and convergence of a new finite volume method for a two sided space fractional diffusion equation in this paper we consider a two sided space fractional diffusion equation with variable coefficients on a finite domain firstly based on the nodal basis functions we present a new fractional finite volume method for the two sided space fractional diffusion equation and derive the implicit scheme and solve it in matrix form secondly we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent finally some numerical examples are given to show the effectiveness of the new numerical method and the results are in excellent agreement with theoretical analysis

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

inhomogeneous dirichlet boundary value problems of space fractional diffusion equations and their finite element approximations we prove the wellposedness of the galerkin weak formulation and petrov galerkin weak formulation for inhomogeneous dirichlet boundary value problems of constant or variable coefficient conservative caputo space fractional diffusion equations we also show that the weak solutions to their riemann liouville analogues do not exist in general in addition we develop an indirect finite element method for the dirichlet boundary value problems of caputo fractional differential equations which reduces the computational work for the numerical solution of variable coefficient fractional diffusion equations from o n 3 to o n and the memory requirement from o n 2 to o n on any quasiuniform space partition we further prove a nearly sharp error estimate for the method which is expressed in terms of the smoothness of the prescribed data of the problem only we carry out numerical experiments to investigate the performance of the method in comparison with the galerkin finite element method

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

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

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and non smooth solutions abstract we consider the initial boundary value problem of the time fractional nonlinear sine gordon equation and the fractional derivative is described in caputo sense with the order 1

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

high order numerical methods for the riesz space fractional advection dispersion equations abstract in this paper we propose high order numerical methods for the riesz space fractional advection dispersion equations rsfade on a finite domain the rsfade is obtained from the standard advection dispersion equation by replacing the first order and second order space derivative with the riesz fractional derivatives of order 0 1 and 1 2 respectively firstly we utilize the weighted and shifted grunwald difference operators to approximate the riesz fractional derivative and present the finite difference method for the rsfade specifically we discuss the crank nicolson scheme and solve it in matrix form secondly we prove that the scheme is unconditionally stable and convergent with the accuracy of o 2 h 2 thirdly we use the richardson extrapolation method rem to improve the convergence order which can be o 4 h 4 finally some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent with theoretical analysis

a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of variable coefficient conservative fractional diffusion equations fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior recently spectral galerkin methods were developed for space fractional diffusion equations aiming at achieving exponential convergence an optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity an optimal order error estimate in the l 2 norm was proved via the well known nitsche lifting technique under the assumption that the true solution to the corresponding boundary value problem of the fractional diffusion equation has the required regularity for each right hand side in this paper we show that the true solution to the dirichlet boundary value problem of a conservative fractional diffusion equation of order 2 with 0 1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional sobolev space h 3 2 in general but is still in the besov space b 3 2 l 2 hence the provable convergence rate of a spectral galerkin method in the l 2 norm is at most of the order o n 3 2 where n is the degree of the polynomial space in the numerical method numerical experiments show that the spectral galerkin method exhibits a subquadratic convergence in the l 2 norm for any 0 1 we develop a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of one sided variable coefficient conservative fractional diffusion equations the method has a proved high order convergence rate of arbitrary order i without requiring the smoothness of the true solution u to the given boundary value problem but only assuming that the diffusivity coefficient and the right hand source term have the desired regularity ii for a variable diffusivity coefficient and iii for an inhomogeneous dirichlet boundary condition numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right hand source term have the desired regularity

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

modelling anomalous diffusion using fractional bloch torrey equations on approximate irregular domains abstract diffusion weighted imaging is an in vivo non invasive medical diagnosis technique that uses the brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment recent work highlights that the diffusion induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay to investigate the underlying mechanism of this deviated signal decay diffusion is re examined through the bloch torrey equation by using fractional calculus with respect to both time and space in this study we explore the influence of the complex geometrical structure on the diffusion process an effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two dimensional time space riesz fractional partial differential equation with dirichlet boundary conditions this scheme is proved to be unconditionally stable and convergent numerical examples are given to support our analysis we then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time space fractional bloch torrey equations on irregular domains