hyperbolic type transport equations abstract in recent years hyperbolic type transport equations have acquired a great deal of importance in problems ranging from theoretical physics to biology in spite of their greater mathematical difficulty as compared with their parabolic type analogs arising from the framework of linear irreversible thermodynamics they have in many ways superseded the latter ones although the use of this type of equations is well known since the last century through the telegraphist equation of electromagnetic theory their use in studying several problems in transport theory is hardly fifty years old in fact the first appearance of a hyperbolic type transport equation for the problem of heat conduction dates back to cattaneos work in 1948 three years later in 1951 s goldstein showed how in the theory of stochastic processes this type of an equation is obtained in the continuous limit of a one dimensional persistent random walk problem after that other phenomenological derivations have been offered for such equations the main purpose of this paper is to critically discuss a derivation of a hyperbolic type fokker planck equation recently presented using the same ideas as m s green did in 1952 to provide the stochastic foundations of irreversible statistical mechanics arguments are given to show that such an equation as well as transport equations derived from it by taking appropriate averages are at most approximate and that a much more detailed analysis is required before asserting their validity

statistical analysis of the music algorithm in the presence of modeling errors taking into account the resolution probability this paper considers the statistical performance of the music method under the condition that two closely spaced sources impinging on an array of sensors are effectively resolved i e the spectrum exhibits two peaks in the neighborhood of the true directions of arrival doa the music algorithm is known to have an infinite resolution power in theory however in the presence of modeling errors sources can not be resolved with certainty even if the array correlation matrix is perfectly known the focus of this paper is to predict the bias and variance of the doa estimates taking into account the possible resolution failure of music this performance prediction based on our recent mathematical investigation is new to the best of our knowledge a general mathematical framework to derive closed form expressions of the bias and variance versus the model mismatch conditioned on a general statistical resolution test is proposed in order to illustrate our mathematical approach statistical tests with one and two conditions respectively are investigated the accuracy of the performance prediction is illustrated in a simulation study it is found that the proposed approach outperforms the classical technique which ignores the possible resolution failure of the music algorithm therefore our results provide better tools for determining the necessary antenna calibration accuracy to achieve some targeted specifications on the estimator performance

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

a systolic implementation of the chan foster rrqr algorithm the rank revealing qr factorization is a useful tool in many signal processing applications since it explicitly yields all the necessary information to solve rank deficient least squares problems and subset selection problems to compute signal and noise subspaces etc the authors present a systolic algorithm for computing a rank revealing qr factorization and consider the performance of the algorithm

viscoelastic subdiffusion from anomalous to normal we study viscoelastic subdiffusion in bistable and periodic potentials within the generalized langevin equation approach our results justify the ultra slow fluctuating rate view of the corresponding bistable non markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells the transition kinetics is asymptotically stretched exponential when the potential barrier v 0 several times exceeds thermal energy k b t v 0 ensuremath sim 2 10 k b t and it cannot be described by the non markovian rate theory nmrt the well known nmrt result approximates however ever better with the increasing barrier height the most probable logarithm of the residence times moreover the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power law exponent of free subdiffusion ensuremath alpha such a potential free subdiffusion is ergodic surprisingly in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit however the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height the time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude all these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

linear degrees of freedom in speech production analysis of cineradio and labio film data and articulatory acoustic modeling the following contribution addresses several issues concerning speech degrees of freedom in french oral vowels stop and fricative consonants based on an analysis of tongue and lip shapes extracted from cineradio and labio films the midsagittal tongue shapes have been submitted to a linear decomposition where some of the loading factors were selected such as jaw and larynx position while four other components were derived from principal component analysis pca for the lips in addition to the more traditional protrusion and opening components a supplementary component was extracted to explain the upward movement of both the upper and lower lips in v production a linear articulatory model was developed the six tongue degrees of freedom were used as the articulatory control parameters of the midsagittal tongue contours and explained 96 of the tongue data variance these control parameters were also used to specify the frontal lip width dimension derived from the labio film front views finally this model was complemented by a conversion model going from the midsagittal to the area function based on a fitting of the midsagittal distances and the formant frequencies for both vowels and consonants

on complete monotonicity of the prabhakar function and non debye relaxation in dielectrics the three parameters mittag leffler function often referred to as the prabhakar function has important applications mainly in physics of dielectrics in describing anomalous relaxation of non debye type this paper concerns with the investigation of the conditions on the characteristic parameters under which the function is locally integrable and completely monotonic these properties are essential for the physical feasibility of the corresponding models in particular the classical havriliak negami model is extended to a wider range of the parameters the problem of the numerical evaluation of the three parameters mittag leffler function is also addressed and three different approaches are discussed and compared numerical simulations are hence used to validate the theoretical findings and present some graphs of the function under investigation

generalized langevin equation with a three parameter mittag leffler noise abstract the relaxation functions for a given generalized langevin equation in the presence of a three parameter mittag leffler noise are studied analytically the results are represented by three parameter mittag leffler functions exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the laplace transform method the asymptotic behavior of the particle in the short and long time limits are found by using the tauberian theorems it is shown that for large times the particle motion is subdiffusive for 1 and superdiffusive for many previously obtained results are recovered due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

direction of arrival estimates in the presence of wavelength gain and phase errors direction of arrival doa estimation using an array of sensors relies on an accurate characterization of the array manifold in the absence of characterization errors established techniques like music can be shown to perform well both theoretically and in simulation however in the presence of unknown sensor and or source characteristics the performance of most methods degrades significantly we consider the problem of estimating gain and phase errors of an array of sensors whose physical positions are known our algorithm assumes that the gain and phase characteristics of the sensors are independent of doa and employs multiple calibration sources with known doa s it differs from other algorithms in that the signal wavelengths are unknown a least squares formulation of the problem is then shown to be np complete implying that an efficient solution is unlikely to exist an implicit enumerative technique is used to obtain the exact solution for the special case of collinear sensors we further show that an inherent ambiguity in the model prevents exact phase characterization unless the wavelength of one calibration source is assumed known a theorem is presented relating the error in doa to the difference between the assumed and true wavelengths of this calibration source simulation results are presented for both noncollinear and collinear arrays

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

pattern formation in a fractional reaction diffusion system abstract we investigate pattern formation in a fractional reaction diffusion system by the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives we further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives as a result we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics so when the order of time derivative increases and approaches the value of 1 5 the special structure formation switches to homogeneous oscillations in the case of space fractional derivatives the decrease of the order of these derivatives leads to more contrast dissipative structures the variational principle is used to find the approximate solution of such fractional reaction diffusion model in addition we provide a detailed analysis of the characteristic dissipative structures in the system under consideration

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

joint 2 d doa estimation and phase calibration for uniform rectangular arrays a precise model of the array response is required to maintain the performance of direction of arrival doa estimation when modeling errors are present or the sensor environment is time varying autocalibration becomes necessary in this paper the problem of phase autocalibration for uniform rectangular array ura geometries is considered for the case with a single source a simple and robust least squares algorithm for joint 2 d doa estimation and phase calibration is presented when performing phase autocalibration with a ura the phase and doa parameters cannot be identified together without ambiguity this problem is discussed and a suitable remedy is suggested an approximate cramer rao bound and analytical expressions for the mean squared error performance of the proposed estimator are presented the proposed algorithm for phase autocalibration is extended for the case with multiple sources the results are evaluated using a representative body of simulations

array shape calibration using sources in unknown locations a maximum likelihood approach sensor location uncertainty can severely degrade the performance of direction finding systems an iterative maximum likelihood method for simultaneously estimating directions of arrival doa and sensor locations is developed to alleviate this problem the case of nondisjoint sources i e sources observed in the same frequency cell and at the same time is emphasized the algorithm converges to the global maximum of the likelihood function if the initial conditions are sufficiently good numerical examples are presented illustrating the performance of the proposed technique a distinctive feature of the algorithm is its ability to locate the sensors accurately without deploying calibration sources at known locations

generalized nonlinear yule models with the aim of considering models related to random graphs growth exhibiting persistent memory we propose a fractional nonlinear modification of the classical yule model often studied in the context of macroevolution here the model is analyzed and interpreted in the framework of the development of networks such as the world wide web nonlinearity is introduced by replacing the linear birth process governing the growth of the in links of each specific webpage with a fractional nonlinear birth process with completely general birth rates among the main results we derive the explicit distribution of the number of in links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction the mean value of the latter distribution is also calculated explicitly in the most general case furthermore in order to show the usefulness of our results we particularize them in the case of specific birth rates giving rise to a saturating behaviour a property that is often observed in nature the further specialization to the non fractional case allows us to extend the yule model accounting for a nonlinear growth

fractional calculus with an integral operator containing a generalized mittag leffler function in the kernel in this paper we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized mittag leffler functions e a b c k z email protected n 0 c k email protected email protected z nn z b email protected c r a max 0 r k 1 r k 0 in its kernel l n being the familiar pochhammer symbol a number of corollaries and consequences of the main results presented here are also considered

on the asymptotic performance analysis of subspace doa estimation in the presence of modeling errors case of music this paper provides a new analytic expression of the bias and rms error root mean square error of the estimated direction of arrival doa in the presence of modeling errors in first order approximations of the rms error are derived which are accurate for small enough perturbations however the previously available expressions are not able to capture the behavior of the estimation algorithm into the threshold region in order to fill this gap we provide a second order performance analysis which is valid in a larger interval of modeling errors to this end it is shown that the doa estimation error for each signal source can be expressed as a ratio of hermitian forms with a stochastic vector containing the modeling error then an analytic expression for the moments of such a hermitian forms ratio is provided finally a closed form expression for the performance bias and rms error is derived simulation results indicate that the new result is accurate into the region where the algorithm breaks down

a sensitivity analysis of the music algorithm a first order analysis of the sensitivity of the music algorithm to system errors which cause differences between the array manifold used by music and the true array manifold is presented the effect of such errors on the direction of arrival estimates is investigated the ability of the music algorithm to resolve two closely spaced sources in the presence of phase and gain errors in the array elements or the receivers or errors in the element locations is analyzed a formula for computing the failure threshold of music as a function of source separation and other system parameters is derived and tested by simulation the analysis assumes that the exact covariance matrix of array element outputs is known the infinite data case

analysis of the asymptotic relative efficiency of the music algorithm an analytical performance evaluation of the errors of the direction of arrival estimates obtained by the music algorithm for uncorrelated sources is provided explicit asymptotic formulas are derived for the means and the covariance of the estimates the covariances are then compared to the cramer rao lower bound it is shown that for a single course the music algorithm is asymptotically efficient for multiple sources the algorithm is not efficient in general however it approaches asymptotic efficiency when the snrs signal to noise ratios of all sources tend to infinity it is illustrated by several test cases that the relative efficiency of the music algorithm is nearly one under a wide range of parameter variations the analytic performance evaluation thus confirms empirical evidence to the excellent performance of the music algorithm for narrowband signals

music for multidimensional spectral estimation stability and super resolution this paper presents a performance analysis of the multiple signal classification music algorithm applied on d dimensional single snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain inspired by the matrix pencil form we construct a d fold hankel matrix from the measurements and exploit its vandermonde decomposition in the noiseless case music amounts to identifying a noise subspace evaluating a noise space correlation function and localizing frequencies by searching the s smallest local minima of the noise space correlation function in the noiseless case 2s d measurements guarantee an exact reconstruction by music as the noise space correlation function vanishes exactly at true frequencies when noise exists we provide an explicit estimate on the perturbation of the noise space correlation function in terms of noise level dimension d the minimum separation among frequencies the maximum and minimum amplitudes while frequencies are separated by 2 rayleigh length rl at each direction as a by product the maximum and minimum non zero singular values of the multidimensional vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes under the 2 rl separation condition if noise is i i d gaussian we show that perturbation of the noise space correlation function decays like sqrt log bf n bf n as the sample size bf n increases when the separation among frequencies drops below 2 rl our numerical experiments show that the noise tolerance of music obeys a power law with the minimum separation of frequencies

performance analysis of time reversal music in this paper we study the performance of multiple signal classification music in computational time reversal tr applications the analysis builds upon classical results on first order perturbation of singular value decomposition the closed form of mean squared error mse matrix of tr music is derived for the single frequency case in both multistatic co located and non co located scenarios the proposed analysis is compared with cramer rao lower bound crlb and it is exploited for comparison of tr music when linear and nonlinear multiple scattering is present finally a numerical analysis is provided to confirm the theoretical findings

semiblind multiuser mimo channel estimation using capon and music techniques channel state information csi is required at the receiver side of multiple input multiple output mimo systems for decoding space time codes in this paper new semiblind techniques are developed to jointly estimate the receiver csi of multiple multiantenna transmitters that use orthogonal space time block codes ostbcs the proposed techniques are based on the extension of the concepts of the popular capon and music spectral estimators to the problem of multiuser channel estimation our methods blindly estimate the subspace that contains the user channel matrices and then use only a few training symbol blocks to extract the user csi from this subspace it is demonstrated that in addition to improving the bandwidth efficiency the proposed techniques offer better channel estimation accuracy as compared with the standard nonblind least squares ls channel estimation method moreover it is shown that using the proposed capon and music based semiblind channel estimators along with the coherent maximum likelihood ml or the minimum variance mv multiuser mimo receivers results in a substantially better performance as compared with the case when the standard ls channel estimates are used

some applications of the fractional poisson probability distribution physical and mathematical applications of the recently invented fractional poisson probability distribution have been presented as a physical application a new family of quantum coherent states has been introduced and studied as mathematical applications we have developed the fractional generalization of bell polynomials bell numbers and stirling numbers of the second kind the appearance of fractional bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states fractional stirling numbers of the second kind have been introduced and applied to evaluate the skewness and kurtosis of the fractional poisson probability distribution function a representation of the bernoulli numbers in terms of fractional stirling numbers of the second kind has been found in the limit case when the fractional poisson probability distribution becomes the poisson probability distribution all of the above listed developments and implementations turn into the well known results of the quantum optics and the theory of combinatorial numbers

on the role of the number of degrees of freedom of the field in mimo channels the aim of this paper is to investigate the relationship between the effective number of degrees of freedom endf of a multiple input multiple output mimo channel and the number of degrees of freedom ndf of the field in particular the ndf of the field represents an upperbound for the endf of a mimo channel consequently the evaluation of the maximum endf can be carried out by adopting the literature on the ndf based on the spatial bandlimitation properties of the field this approach allows a simple and intuitive explanation of the maximum endf in a mimo channel based on a geometrical interpretation of the spatial channel in terms of spatial nyquist intervals

consistency and mse performance of music based doa of a single source in white noise with randomly missing data music is a popular algorithm for estimating the direction of arrival doa in array signal processing applications in this paper we analyze the performance of the music algorithm for a single source system in the presence of noisy and missing data when only a random subset of the entries in the data matrix are observed we prove consistency of the doa estimate when signal from a single source is impinging on low coherence arrays and derive an analytic expression for the mean squared error mse performance of music for the case of uniform linear arrays in the large array and relatively large sample setting our analysis is mathematically justified in both the sample rich and deficient regimes the expression for the mse is a simple function of array geometry signal to noise ratio snr the fraction of entries observed and the ratio of the number of sensors to number of snapshots we derive a phase transition threshold which separates a regime where music is consistent from a regime where music is inconsistent this threshold depends upon the snr the probability of observing entries in the data matrix and number of sensors and snapshots in a simple manner which we make explicit

performance analysis of the mvdr spatial spectrum estimator the performance of the minimum variance distortionless response spectrum estimator is analyzed finite data effects and the sensitivity of the method to random perturbations in the signal model and in the noise covariance matrix are studied the snapshots are assumed to be complex independent identically distributed gaussian vectors an expression for the exact model asymptotic bias is also derived analytical expressions for the variance and the bias of the estimator are derived and compared with simulation results these expressions are then employed to study the characteristics of the estimator

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

estimation of sensor gain and phase the problem of estimating the direction independent gain and phase characteristics of an array of sensors using knowledge of the true field covariance at the sensor locations is considered a concise expression for the log likelihood function is derived and several mathematical properties of this objective function are given the cramer rao c r lower bounds on the variances of gain and phase estimates are derived with the plane wave in isotropic noise considered as a special case the maximum likelihood estimates are shown to be consistent asymptotically efficient and asymptotically normal a simple estimator is proposed which is consistent and which gives good initial estimates for a newton algorithm for finding the maximum likelihood solution comparison of the maximum likelihood estimates and the c r bounds is given for the plane wave in isotropic noise example

effect of spatial smoothing on the performance of subspace methods in the presence of array model errors ln this paper the effect of spatial smoothing forward smoothing and forward backward smoothing on the performance of unweighted and weighted subspace methods in the presence of array model errors for direction of arrival doa estimation is studied theoreti cal expressions for the mean squared error mse in doa are obtained based on a common framework of analysis for weighted subspace methods optimal weighting matrices are presented which minimize the mse in doa two typical errors random errors and gain and phase errors are considered as examples for illustration simulations are carried out to substantiate the theory developed for the cases considered smoothing improves the performance of esprit and minimum norm method while it is not so for music forward backward smoothing improves the per formance of all methods compared to that of forward smoothing

application of cumulants to array signal processing v sensitivity issues for pt iv see ibid vol 45 p 2265 76 1997 virtual esprit vespa imposes a slight constraint on the array i e only a pair of identical sensors are required we follow the classical procedure to define the sensitivity of vespa with respect to each model parameter and derive closed forms for the sensitivity formulas we justify our derivations by comparing the theoretical formulas with the results from monte carlo simulations additionally we demonstrate by simulations that vespa is more robust to model mismatches than esprit in most situations

applications of cumulants to array processing i aperture extension and array calibration an interpretation for the use of cumulants in narrowband array processing problems is proposed it is shown how fourth order cumulants of multichannel observations increase the directional information compared with second order statistics based on the interpretation it is shown how cumulants can be used to increase the effective aperture of an arbitrary antenna array the amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far field sources is also investigated it is proven that the presence of a doublet and use of fourth order cumulants is sufficient to accomplish this task the proposed approach is computationally efficient and more general than covariance based algorithms that have addressed the calibration problem under constraints a class of beamforming techniques is proposed to recover the source waveforms proposed estimation procedures are based on cumulants which bring insensitivity to the spatial correlation structure of additive gaussian measurement noise simulations are provided to illustrate the use of the proposed algorithms

evolution of the initial box signal for time fractional diffusion wave equation in a case of different spatial dimensions abstract in the case of time fractional diffusion wave equation considered in the spatial domain x evolution of initial box signal was investigated by mainardi f mainardi fractional relaxation oscillation and fractional diffusion wave phenomena chaos solitons fractals 7 1996 1461 1477 in the present paper we supplement mainardi s results with additional numerical calculations illustrating the behavior of the solution and solve the corresponding problems for axisymmetric and central symmetric cases the obtained results show an unusual behavior of solutions

some uniqueness and existence results for the initial boundary value problems for the generalized time fractional diffusion equation in this paper some uniqueness and existence results for the solutions of the initial boundary value problems for the generalized time fractional diffusion equation over an open bounded domain gx 0 t email protected r n are given to establish the uniqueness of the solution a maximum principle for the generalized time fractional diffusion equation is used in turn the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is considered in the paper too another important consequence of the maximum principle is the continuous dependence of the solution on the problem data to show the existence of the solution the fourier method of the variable separation is used to construct a formal solution under certain conditions the formal solution is shown to be a generalized solution of the initial boundary value problem for the generalized time fractional diffusion equation that turns out to be a classical solution under some additional conditions

large system spectral analysis of covariance matrix estimation eigendecomposition of estimated covariance matrices is a basic signal processing technique arising in a number of applications including direction of arrival estimation power allocation in multiple input multiple output mimo transmission systems and adaptive multiuser detection this paper uses the theory of non crossing partitions to develop explicit asymptotic expressions for the moments of the eigenvalues of estimated covariance matrices in the large system asymptote as the vector dimension and the dimension of signal space both increase without bound while their ratio remains finite and nonzero the asymptotic eigenvalue distribution is also obtained from these eigenvalue moments and the stieltjes transform and is extended to first order approximation in the large sample size limit numerical simulations are used to demonstrate that these asymptotic results provide good approximations for finite systems of moderate size

weighted subspace fitting for general array error models model error sensitivity is an issue common to all high resolution direction of arrival estimators much attention has been directed to the design of algorithms for minimum variance estimation taking only finite sample errors into account approaches to reduce the sensitivity due to array calibration errors have also appeared in the literature herein one such approach is adopted that assumes that the errors due to finite samples and model errors are of comparable size a weighted subspace fitting method for very general array perturbation models is derived this method provides minimum variance estimates under the assumption that the prior distribution of the perturbation model is known interestingly the method reduces to the wsf mode estimator if no model errors are present vice versa assuming that model errors dominate the method specializes to the corresponding model errors only subspace fitting method unlike previous techniques for model errors the estimator can be implemented using a two step procedure if the nominal array is uniform and linear and it is also consistent even if the signals are fully correlated the paper also contains a large sample analysis of one of the alternative methods namely mapprox it is shown that mapprox also provides minimum variance estimates under reasonable assumptions

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

efficient quadratic detection in perturbed arrays via fourier transform techniques matched field beamforming used in combination with a generalized likelihood ratio test is the most common detector structure in array processing situations unfortunately various array perturbations caused by phase calibration propagation effects or modeling errors can cause the sensor observations to become only partially correlated limiting the performance of traditional matched field beamformers that assume perfect coherence of the signal wavefronts quadratic array processing is optimal for many perturbed array problems however direct implementation poses a significant computational burden we show that under certain conditions the optimal quadratic detector for dealing with perturbed arrays can be approximately realized efficiently and robustly employing only discrete fourier transforms to deal with spatial processing in addition we show that the proposed spatial processing allows for convenient integration of conventional frequency domain methods for angle of arrival searches our proposed array detection structure provides the robustness and performance benefits of complicated quadratic processing at a computational cost comparable with that of traditional matched field beamforming

direction of arrival estimation using rank revealing qr factorization the authors describe a novel technique for direction of arrival estimation based on computing a permutation matrix e and a qr factorization re hb of the permuted covariance matrix r such that a possible rank deficiency of r is revealed in the triangular factor b having a minimum norm lower right block a subset of the columns of the orthogonal matrix h is shown to be orthogonal to the direction vectors of sources and hence can be used to estimate their bearings the cost of this algorithm is only slightly more than that of one qr factorization but is much lower than that of an eigen decomposition simulation results are included to show that the proposed method performs nearly as well as music in terms of signal resolution bias and variance of the estimated bearings

analysis of facial motion patterns during speech using a matrix factorization algorithm this paper presents an analysis of facial motion during speech to identify linearly independent kinematic regions the data consists of three dimensional displacement records of a set of markers located on a subject s face while producing speech a qr factorization with column pivoting algorithm selects a subset of markers with independent motion patterns the subset is used as a basis to fit the motion of the other facial markers which determines facial regions of influence of each of the linearly independent markers those regions constitute kinematic eigenregions whose combined motion produces the total motion of the face facial animations may be generated by driving the independent markers with collected displacement records

an analysis of esprit under random sensor uncertainties a general expression for the mean square error mse of the esprit direction of arrival doa estimation of narrowband far field sources under random sensor perturbations is derived explicit solutions for the case of an arbitrary esprit array geometry with one and two sources are given solutions for the case of a uniform linear array and arbitrary number of sources are found provided that the number of sensors is large for this case it is found that the mse of esprit with maximum aperture i e maximum overlapping subarrays is lower than that of esprit with nonoverlapping subarrays a comparison with music suggests that the mse for music is lower than that for esprit furthermore for the cases studied it is shown that the mse of esprit depends only on sensor phase errors while that of music is dependent on both sensor grain and phase errors

a weighted numerical algorithm for two and three dimensional two sided space fractional wave equations a weighed numerical scheme that can effectively solve the fractional wave equation in a finite domain is provided we focus on detailedly discussing the two and three dimensional two sided space fractional wave equations with homogeneous boundary conditions a second order finite difference scheme is used to discretize the space fractional derivative and the time derivative additionally the numerical results confirm that the weighted numerical algorithm is convergent with second order accuracy in both space and time directions for the homogeneous boundary fractional problems

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

multiple invariance esprit a subspace fitting formulation of the esprit problem is presented that provides a framework for extending the algorithm to exploit arrays with multiple invariances in particular a multiple invariance mi esprit algorithm is developed and the asymptotic distribution of the estimates is obtained simulations are conducted to verify the analysis and to compare the performance of mi esprit with that of several other approaches the excellent quality of the mi esprit estimates is explained by recent results which state that under certain conditions subspace fitting methods of this type are asymptotically efficient

time fractional particle deposition in porous media in the percolation process where fluids carry small solid particles particle deposition causes a real time permeability change of the medium as the swarm of particles propagates along the medium then the permeability change influences percolation and deposition behaviors as a feedback this fact triggers memory effect in the deposition dynamics which means the particulate transport and deposition behaviors become history dependent in this paper we conduct the time fractional generalization of the classical phenomenological model of particle deposition in porous media to incorporate the memory effect we tested and compared the effects of employing different types of fractional operators i e the riemann liouville type the hadamard type and the prabhakar type numerical simulation results show that the system behaviors vary according to the change of distinct memory kernels in an expected way we then discuss the physical meaning of the time fractional generalization it is shown that different types of fractional operators unanimously ground themselves on the local newtonian time transformation in a complex system which is equivalent to a class of history integrals by the introduction of various memory kernels it enables the model to more powerfully fit and approximate observed data further the fundamental meaning of this work is not to show which fractional operator is better but to argue collectively the legitimacy and practicality of a non markovian particle deposition dynamics in porous media and in fact it is admissible to a bunch of memory kernels which differ greatly from each other in functional forms hopefully the presented generalized mass conservation formalism offers a broader framework to investigate transport problems in porous media

two dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time fractional diffusion equation the theory of diffusive stresses based on the time fractional diffusion equation is formulated the source problem is discussed as well as the cauchy problem the stresses are found in axially symmetric cases for plane deformation the numerical results for the concentration and stress distributions are presented graphically for various values of order of fractional derivative

r dimensional esprit type algorithms for strictly second order non circular sources and their performance analysis high resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second order so noncircular nc sources allow for a lower estimation error and can resolve twice as many sources in this paper we derive the r mathchar 702dd nc standard esprit and the r mathchar 702dd nc unitary esprit algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals they are applicable to shift invariant r mathchar 702dd antenna arrays and do not require a centro symmetric array structure moreover we present a first order asymptotic performance analysis of the proposed algorithms which is based on the error in the signal subspace estimate arising from the noise perturbation the derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal to noise ratio snr i e the results become exact for either high snrs or a large sample size we also provide mean squared error mse expressions where only the assumptions of a zero mean and finite so moments of the noise are required but no assumptions about its statistics are necessary as a main result we analytically prove that the asymptotic performance of both r mathchar 702dd nc esprit type algorithms is identical in the high effective snr regime finally a case study shows that no improvement from strictly non circular sources can be achieved in the special case of a single source

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

glrt based detection estimation for undersampled training conditions for sensors where the number of available independent identically distributed training samples t is less than the number of antenna array elements m we propose nondegenerate properly normalized likelihood ratio lr tests both standard and scale invariant to support detection estimation of m point sources m t in white noise based on a generalized likelihood ratio test glrt approach we demonstrate that these tests can detect music specific ldquooutliersrdquo in the direction of arrival doa estimation of closely spaced independent sources caused by insufficient training volume and or signal to noise ratio snr we then compare the performance of the introduced lrs to other test statistics available in this undersampled regime we show that a search for solutions that increase the introduced lr allows us to replace the detected outliers by proper doa estimates this ldquopredict and curerdquo process leverages the snr ldquogaprdquo between music breakdown and breakdown of maximum likelihood estimation itself the resultant lr maximization makes the associated covariance model statistically ldquoas likelyrdquo as the true covariance matrix and removes the vast percentage of outliers in certain scenarios

hilfer prabhakar derivatives and some applications abstract we present a generalization of hilfer derivatives in which riemann liouville integrals are replaced by more general prabhakar integrals we analyze and discuss its properties furthermore we show some applications of these generalized hilfer prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations and in difference differential equations governing the dynamics of generalized renewal stochastic processes

generalized distributed order diffusion equations with composite time fractional derivative abstract in this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the fourier laplace transform method we represent solutions in terms of infinite series in fox h functions the fractional and second moments are derived by using mittag leffler functions we observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives generalized uniformly distributed order diffusion equation as a model for strong anomalous behavior is analyzed by using tauberian theorem some previously obtained results are special cases of those presented in this paper

fractional diffusion equation and green function approach exact solutions abstract we investigate the solutions of a fractional diffusion equation with radial symmetry by using the green function approach and by taking the n dimensional case into account in our analysis a spatial time dependent diffusion coefficient is considered i e d r t d t 1 r the presence of external forces f r k r e with e 1 and f r kr k r e is also taken into account in particular we discuss the results obtained by employing boundary conditions defined on a finite interval and afterwards the analysis is extended to a semi infinite interval finally we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work

the resolution threshold of a direction finding algorithm for diversely polarized arrays subspace based algorithms for narrowband direction of arrival doa estimation require detailed knowledge of the array response the array manifold and assume that the noise covariance matrix is known up to a scaling factor in practice these quantities are not known precisely resolution and estimation accuracy can degrade significantly when the array response or the noise covariance deviate from their nominal values we examine the resolution threshold of a recently proposed subspace based algorithm for direction finding with diversely polarized arrays we study finite sample effects and the effects of modeling errors errors in the array manifold or the noise covariance on the resolution threshold a comparison is made between the resolution thresholds of the music algorithm for uniformly polarized arrays and the proposed algorithm for diversely polarized arrays

fractional diffusion equation with a generalized riemann liouville time fractional derivative in this paper the solution of a fractional diffusion equation with a hilfer generalized riemann liouville time fractional derivative is obtained in terms of mittag leffler type functions and fox s h function the considered equation represents a quite general extension of the classical diffusion heat conduction equation the methods of separation of variables laplace transform and analysis of the sturm liouville problem are used to solve the fractional diffusion equation defined in a bounded domain by using the fourier laplace transform method it is shown that the fundamental solution of the fractional diffusion equation with a generalized riemann liouville time fractional derivative defined in the infinite domain can be expressed via fox s h function it is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the caputo and riemann liouville sense are special cases of those diffusion equations with the hilfer generalized riemann liouville time fractional derivative the asymptotic behaviour of the solutions are found for large values of the spatial variable the fractional moments of the fundamental solution of the fractional diffusion equation are obtained the obtained results are relevant in the context of glass relaxation and aquifer problems

performance degradation of doa estimators due to unknown noise fields a statistical performance analysis of subspace based directions of arrival doa estimation algorithms in the presence of correlated observation noise with unknown covariance is presented the analysis of five different estimation algorithms is unified by a single expression for the mean squared doa estimation error which is derived using a subspace perturbation expansion the analysis assumes that only a finite amount of array data is available

fractional langevin equation overdamped underdamped and critical behaviors the dynamical phase diagram of the fractional langevin equation is investigated for a harmonically bound particle it is shown that critical exponents mark dynamical transitions in the behavior of the system four different critical exponents are found i ensuremath alpha c 0 402 ifmmode pm else textpm fi 0 002 marks a transition to a nonmonotonic underdamped phase ii ensuremath alpha r 0 441 dots marks a transition to a resonance phase when an external oscillating field drives the system and iii ensuremath alpha ensuremath chi 1 0 527 dots and iv ensuremath alpha ensuremath chi 2 0 707 dots mark transitions to a double peak phase of the loss when such an oscillating field present as a physical explanation we present a cage effect where the medium induces an elastic type of friction phase diagrams describing over and underdamped regimes with or without resonances show behaviors different from normal

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

min norm interpretations and consistency of music mode and ml the multiple signal characterization music approach its generalization to correlated signals known as the method of direction estimation mode and the deterministic maximum likelihood ml approach for bearing estimation in array processing are shown to be signal subspace fitting approaches in a minimum norm sense mode for example is shown to be an approach in which the array manifold is linearly estimated from principal empirical eigenvectors in a minimum weighted frobenius norm sense using the min norm interpretations a unified proof for strong consistency of the three approaches is provided for stationary and ergodic signals

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

analytical performance assessment of multi dimensional matrix and tensor based esprit type algorithms in this paper we present a generic framework for the asymptotic performance analysis of subspace based parameter estimation schemes it is based on earlier results on an explicit first order expansion of the estimation error in the signal subspace obtained via an svd of the noisy observation matrix we extend these results in a number of aspects firstly we demonstrate that an explicit first order expansion of the higher order svd hosvd based subspace estimate can be derived secondly we show how to obtain explicit first order expansions of the estimation error of arbitrary esprit type algorithms and provide the expressions for r d standard esprit r d unitary esprit r d standard tensor esprit as well as r d unitary tensor esprit thirdly we derive closed form expressions for the mean square error mse and show that they only depend on the second order moments of the noise hence to apply this framework we only need the noise to be zero mean and possess finite second order moments additional assumptions such as gaussianity or circular symmetry are not needed

subset selection for matrices abstract we consider the question of how to delete m k rows from a matrix x r m n so that the resulting matrix a r k n is as non singular as possible bounds for the singular values of a are derived which decrease only algebraically with m and n in addition a number of applications where subset selection is necessary are examined

some results for a fractional diffusion equation with radial symmetry in a confined region abstract we investigate an n dimensional fractional diffusion equation with radial symmetry by taking a spatial and time dependent diffusion coefficient into account i e d r t d t r with d t d t d t the equation is considered in a confined region and subjected to time dependent boundary conditions which may be related to inhomogeneous characteristics of the surfaces confining the system the results show an anomalous spreading of the solutions and an unusual behavior of the survival probability

on the resolution probability of music in presence of modeling errors the problem of resolving closely spaced signal sources using an antenna array remains a difficult one although several estimation methods are available in the literature when the array correlation matrix is known the resolution capability of subspace algorithms is infinitely high however in the presence of modeling errors the resolution deteriorates even for a known correlation matrix in this paper we analyze the music method by way of three different definitions of the resolution assuming gaussian circular random modeling errors we determine the corresponding expressions of the probability of source resolution versus the model mismatch a first series of simulations validates the mathematical expression of the three resolution probabilities a second series of simulations is used to select among them the tightest one to the empirical one the results are useful e g for determining the necessary antenna calibration accuracy to achieve a target performance