on bounded positive stationary solutions for a nonlocal fisher kpp equation abstract we study the existence of stationary solutions for a nonlocal version of the fisher kolmogorov petrovskii piscounov fkpp equation the main motivation is a recent study by berestycki et al 2009 where the nonlocal fkpp equation has been studied and it was shown for the spatial domain r and sufficiently small nonlocality that there are only two bounded non negative stationary solutions here we provide a similar result for r d using a completely different approach in particular an abstract perturbation argument is used in suitable weighted sobolev spaces one aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality i e to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not

self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

generalized bayesian information criterion for source enumeration in array processing we investigate the problem of enumerating source signals impinging on an array of sensors in an information theoretic framework the conventional bayesian information criterion bic does not yield satisfactory performance for this problem because it only considers the density of the observations in order to remedy the limitations of the bic we propose a generalized bayesian information criterion gbic rule by incorporating the density of the sample eigenvalues or corresponding statistics such a density contains extra information and complements the density of the observations in constructing the gbic as a result two different expressions for the gbic are suggested simulation results validate the superiority of the proposed gbic over the conventional bic in terms of correctly determining the number of sources while their computational costs are comparable

the control set of a linear control system on the two dimensional lie group abstract in this paper we explicitly calculate the control sets associated with a linear control system on the two dimensional solvable lie group we show that a linear control system of such kind admits exactly one control set or infinite control sets depending on some algebraic conditions

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

quenching of flames by fluid advection we consider a simple scalar reaction advection diffusion equation with ignitiontype nonlinearity and discuss the following question what kinds of velocity profiles are capable of quenching any given flame provided the velocity s amplitude is adequately large even for shear flows the answer turns out to be surprisingly subtle if the velocity profile changes in space so that it is nowhere identically constant or if it is identically constant only in a region of small measure then the flow can quench any initial data but if the velocity profile is identically constant in a sizable region then the ensuing flow is incapable of quenching large enough flames no matter how much larger the amplitude of this velocity is the constancy region must be wider across than a couple of laminar propagating front widths the proof uses a linear pde associated to the nonlinear problem and quenching follows when the pde is hypoelliptic the techniques used allow the derivation of new nearly optimal bounds on the speed of traveling wave solutions c 2001 john wiley sons inc

a rate of convergence result for the largest eigenvalue of complex white wishart matrices it has been recently shown that if x is an n n matrix whose entries are i i d standard complex gaussian and l1 is the largest eigenvalue of x x there exist sequences mn n and sn n such that l1 mn n sn n converges in distribution to w2 the tracy widom law appearing in the study of the gaussian unitary ensemble this probability law has a density which is known and computable the cumulative distribution function of w2 is denoted f2 in this paper we show that under the assumption that n n 0 we can find a function m continuous and nonincreasing and sequences n n and n n such that for all real s0 there exists an integer n s0 for which if n n n s0 we have with ln n l1 n n n n s s0 n n 2 3 p ln n s f2 s m s0 exp s the surprisingly good 2 3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution w2 is a good approximation to the empirical distribution of ln n in simulations an important fact from the point of view of e g statistical applications

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

effective dimensions and chemical reactions in fluid flows we show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects the stirring protocol of the flow the local properties of the reaction and the global folding dynamics which also depends on the geometry of the container the essence of each of these components can be described by simple functional relations in an ordinary differential equation approach they determine a new chemical rate equation for the concentration which turns out to be coupled to the dynamics of an effective fractal dimension the theory predicts an exponential convergence to the asymptotic chemical state this holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics in the exponential case the theory applies to flows of chaotic time dependence chaotic flows as well

large dimensional behavior of regularized maronna s m estimators of covariance matrices robust estimators of large covariance matrices are considered comprising regularized linear shrinkage modifications of maronna s classical m estimators these estimators provide robustness to outliers while simultaneously being well defined when the number of samples does not exceed the number of variables by applying tools from random matrix theory we characterize the asymptotic performance of such estimators when the numbers of samples and variables grow large together in particular our results show that when outliers are absent many estimators of the regularized maronna type share the same asymptotic performance and for these estimators we present a data driven method for choosing the asymptotically optimal regularization parameter with respect to a quadratic loss robustness in the presence of outliers is then studied in the nonregularized case a large dimensional robustness metric is proposed and explicitly computed for two particular types of estimators exhibiting interesting differences depending on the underlying contamination model the impact of outliers in regularized estimators is then studied with interesting differences with respect to the nonregularized case leading to new practical insights on the choice of particular estimators

model order selection based on information theoretic criteria design of the penalty information theoretic criteria itc have been widely adopted in engineering and statistics for selecting among an ordered set of candidate models the one that better fits the observed sample data the selected model minimizes a penalized likelihood metric where the penalty is determined by the criterion adopted while rules for choosing a penalty that guarantees a consistent estimate of the model order are known theoretical tools for its design with finite samples have never been provided in a general setting in this paper we study model order selection for finite samples under a design perspective focusing on the generalized information criterion gic which embraces the most common itc the theory is general and as case studies we consider a the problem of estimating the number of signals embedded in additive white gaussian noise awgn by using multiple sensors b model selection for the general linear model glm which includes e g the problem of estimating the number of sinusoids in awgn the analysis reveals a trade off between the probabilities of overestimating and underestimating the order of the model we then propose to design the gic penalty to minimize underestimation while keeping the overestimation probability below a specified level for the considered problems this method leads to analytical derivation of the optimal penalty for a given sample size a performance comparison between the penalty optimized gic and common aic and bic is provided demonstrating the effectiveness of the proposed design strategy

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

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

controllability of linear systems on lie groups with finite semisimple center this paper studies controllability for a given linear system sigma on a connected lie group g by taking into consideration the eigenvalues of an associated derivation mathcal d if we assume that the lie group g has finite center and for some tau 0 the identity element of g is an interior point of its reachable set at time tau then the system is controllable if mathcal d has only eigenvalues with zero real part

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

instability in reaction superdiffusion systems we study the effect of superdiffusion on the instability in reaction diffusion systems it is shown that reaction superdiffusion systems close to a turing instability are equivalent to a time dependent ginzburg landau model and the corresponding free energy is introduced this generalized free energy which depends on the superdiffusion exponent governs the stability dynamics and the fluctuations of reaction superdiffusion systems near the turing bifurcation in addition we show that for a general n component reaction superdiffusion system a fractional complex ginzburg landau equation emerges as the amplitude equation near a hopf instability numerical simulations of this equation are carried out to illustrate the effect of superdiffusion on spatiotemporal patterns finally the effect of superdiffusion on the instability in the brusselator model as a special case of reaction diffusion systems is studied in general superdiffusion introduces a new parameter that changes the behavior of the system near the instability

stieltjes integral representation of effective diffusivities in time dependent flows a stieltjes integral representation for the effective diffusivity of a passive scalar in time dependent incompressible flows is developed the representation provides a summability formula for the perturbative expansion of the diffusivity in powers of the p eclet number in particular upper and lower bounds on the effective diffusivity are obtained from pad e approximants of the series

an mdl framework for sparse coding and dictionary learning the power of sparse signal modeling with learned overcomplete dictionaries has been demonstrated in a variety of applications and fields from signal processing to statistical inference and machine learning however the statistical properties of these models such as underfitting or overfitting given sets of data are still not well characterized in the literature as a result the success of sparse modeling depends on hand tuning critical parameters for each data and application this work aims at addressing this by providing a practical and objective characterization of sparse models by means of the minimum description length mdl principle a well established information theoretic approach to model selection in statistical inference the resulting framework derives a family of efficient sparse coding and dictionary learning algorithms which by virtue of the mdl principle are completely parameter free furthermore such framework allows to incorporate additional prior information to existing models such as markovian dependencies or to define completely new problem formulations including in the matrix analysis area in a natural way these virtues will be demonstrated with parameter free algorithms for the classic image denoising and classification problems and for low rank matrix recovery in video applications however the framework is not limited to this imaging data and can be applied to a wide range of signal and data types and tasks

analysis of the performance and sensitivity of eigendecomposition based detectors a new framework is presented for the analysis of the performance of detection methods such as aic and mdl which are based on the eigenvalues of the sample covariance matrix it is shown that theoretical analysis of the probabilities of overestimation and underestimation can be much more conveniently carried out via a proposed particularly simple sequence of statistics also the breakdown of these detection methods in the presence of model nonidealities is explored by theory simulations and experimentation with real array data for example theoretical arguments are given to demonstrate the high degree of sensitivity of the detectors to unknown deviations of the noise from whiteness

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

nonparametric detection of signals by information theoretic criteria performance analysis and an improved estimator determining the number of sources from observed data is a fundamental problem in many scientific fields in this paper we consider the nonparametric setting and focus on the detection performance of two popular estimators based on information theoretic criteria the akaike information criterion aic and minimum description length mdl we present three contributions on this subject first we derive a new expression for the detection performance of the mdl estimator which exhibits a much closer fit to simulations in comparison to previous formulas second we present a random matrix theory viewpoint of the performance of the aic estimator including approximate analytical formulas for its overestimation probability finally we show that a small increase in the penalty term of aic leads to an estimator with a very good detection performance and a negligible overestimation probability

reaction spreading on graphs we study reaction diffusion processes on graphs through an extension of the standard reaction diffusion equation starting from first principles we focus on reaction spreading i e on the time evolution of the reaction product m t at variance with pure diffusive processes characterized by the spectral dimension d s the important quantity for reaction spreading is found to be the connectivity dimension d l numerical data in agreement with analytical estimates based on the features of n independent random walkers on the graph show that m t t d l in the case of erdos renyi random graphs the reaction product is characterized by an exponential growth m t e t with proportional to ln k where k is the average degree of the graph

on information theoretic criteria for determining the number of signals in high resolution array processing an important problem in high resolution array processing is the determination of the number of signals arriving at the array information theoretic criteria provide a means to achieve this two commonly used criteria are the akaike information criterion aic and minimum descriptive length mdl criterion while the aic tends to overestimate even at a high signal to noise ratio snr the mdl criterion tends to underestimate at low or moderate snr by excluding irrelevant parameters a new log likelihood function has been chosen utilizing this new log likelihood function gives a set of more accurate estimates of the eigenvalues and in the establishment of modified information theoretic criteria which moderate the performance of the aic and the mdl criterion computer simulations confirm that the modified criteria have superior performance

on the behavior of information theoretic criteria for model order selection the akaike 1974 information criterion aic and the minimum description length mdl are two well known criteria for model order selection in the additive white noise case our aim is to study the influence on their behavior of a large gap between the signal and the noise eigenvalues and of the noise eigenvalue dispersion our results are mostly qualitative and serve to explain the behavior of the aic and the mdl in some cases of great practical importance we show that when the noise eigenvalues are not clustered sufficiently closely then the aic and the mdl may lead to overmodeling by ignoring an arbitrarily large gap between the signal and the noise eigenvalues for fixed number of data samples overmodeling becomes more likely for increasing the dispersion of the noise eigenvalues for fixed dispersion overmodeling becomes more likely for increasing the number of data samples undermodeling may happen in the cases where the signal and the noise eigenvalues are not well separated and the noise eigenvalues are clustered sufficiently closely we illustrate our results by using simulations from the effective channel order determination area

pattern formation by super diffusion in fitzhugh nagumo model abstract the aim of this paper is to examine the turing instability and pattern formation in the fitzhugh nagumo model with super diffusion in two dimensional numerical simulation we also studied the effects of the super diffusive exponent on pattern formation concluding that with the presence of super diffusion the stable homogenous steady state becomes unstable by using the stability analysis of local equilibrium point we procure the conditions which ensure that the turing and hopf bifurcations occur for pattern selection the weak nonlinear multi scale analysis is used to derive the amplitude equations of the stationary patterns we then apply amplitude equations and observe that this model has very rich dynamical behaviors such as stripes spots and hexagon patterns the complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation the simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically

lattice boltzmann simulation of pattern formation under cross diffusion in this paper a lattice boltzmann model for cross reaction diffusion system is proposed in order to recover the cross diffusion equations by lattice boltzmann method we transform the equations into a complex reaction diffusion equation this complex reaction diffusion equation is recovered with higher order accuracy of the truncation error based on this model two typical reaction diffusion systems with cross diffusion predator prey system and inhomogeneous brusselator model are simulated the numerical results show that this model can be used to simulate the cross reaction diffusion systems

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

spatiotemporal dynamics induced by delay and diffusion in a predator prey model with mutual interference among the predator abstract in this work we study a reaction diffusion predator prey model with mutual interference among the predators while searching for food we prove that the model exhibits bistability which indicates that there are no patterns for our model when time delay is incorporated into the model multiple stability switches phenomenon of positive constant steady state emerged by taking delay as a bifurcation parameter the hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay the algorithm for determining the direction and the stability of the bifurcating periodic solutions is also derived the delay diffusion driven turing instability of the positive constant steady state is investigated our results show that delay and diffusion can create periodic oscillatory patterns of spatially homogeneous and inhomogeneous and turing patterns

simple and efficient nonparametric method for estimating the number of signals without eigendecomposition inspired by the computational simplicity and numerical stability of qr decomposition a nonparametric method for estimating the number of signals without eigendecomposition mense is proposed for the coherent narrowband signals impinging on a uniform linear array ula by exploiting the array geometry and its shift invariance property to decorrelate the coherency of signals through subarray averaging the number of signals is revealed in the rank of the qr upper trapezoidal factor of the autoproduct of a combined hankel matrix formed from the cross correlations between some sensor data since the infection of additive noise is defused signal detection capability is improved a new detection criterion is then formulated in terms of the row elements of the qr upper triangular factor when finite array data are available and the number of signals is determined as a value of the running index for which this ratio criterion is maximized where the qr decomposition with column pivoting is also used to improve detection performance the statistical analysis clarifies that the mense detection criterion is asymptotically consistent furthermore the proposed mense algorithm is robust against the array uncertainties including sensor gain and phase errors and mutual coupling and against the deviations from the spatial homogeneity of noise model the effectiveness of the mense is verified through numerical examples and the simulation results show that the mense is superior in detecting closely spaced signals with a small number of snapshots and or at relatively low signal to noise ratio snr

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

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

number of source signal estimation by the mean squared eigenvalue error detection of the number of source signals noss in the presence of additive noise is considered we present a new approach denoted by the mean squared eigenvalue error msee the msee is the mean squared error between the desired noise free eigenvalues and the available estimated eigenvalues the approach investigates and analyzes the probabilistic distribution of the available eigenvalue estimates and revisits proper thresholding of these sorted values the optimum noss is provided by minimizing the msee a probabilistic worst case technique is proposed to estimate the value of the msee by using only the available data it is shown that the proposed method is consistent as the data length increases it is also shown that the method is consistent as the signal to noise ratio snr increases simulation results illustrate advantages of the msee over competing approaches and confirm effectiveness and robustness of the msee even in low snr or small sample size scenarios

detection of the number of coherent signals by the mdl principle an approach is presented to the problem of detecting the number of sources impinging on a passive sensor array that is based on j rissanen s 1983 minimum description length mdl principle the approach is applicable to any type of sources including the case of sources which are fully correlated referred to as the coherent signals case two slightly different detection criteria are derived both requiring the estimation of the locations of the sources the first is tailored to the detection problem per se whereas the second is tailored to the combined detection estimation problem consistency of the two criteria is proved and their performance is demonstrated by computer simulations

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

multifractal random walk in copepod behavior abstract a 3d copepod trajectory is recorded in the laboratory using two digital cameras the copepod undergoes a very structured type of trajectory with successive moves displaying intermittent amplitudes we perform a statistical analysis of this 3d trajectory using statistical tools developed in the field of turbulence and anomalous diffusion in natural sciences we show that the walk belongs to multifractal random walks characterized by a nonlinear moment scaling function for the distance versus time to our knowledge this is the first experimental study of multifractal anomalous diffusion in natural sciences we then propose a new type of stochastic process reproducing these multifractal scaling properties this can be directly used for stochastic numerical simulations and is thus of important potential applications in the field of animal movement study and more generally of anomalous diffusion studies

an information theoretic approach to source enumeration in array signal processing in this paper a new information theoretic algorithm is proposed for signal enumeration in array processing the approach is based on predictive description length pdl that is defined as the length of a predictive code for the set of observations we assume that several models with each model representing a certain number of sources will compete the pdl criterion is computed for the candidate models and is minimized over all models to select the best model and to determine the number of signals in the proposed method the correlation matrix is decomposed into two orthogonal components in the signal and noise subspaces the maximum likelihood ml estimates of the angles of arrival are used to find the projection of the sample correlation matrix onto the signal and noise subspaces the summation of the ml estimates of these matrices is the ml estimate of the correlation matrix this method can detect both coherent and noncoherent signals the proposed method can be used online and can be applied to time varying systems and target tracking

controllability of linear systems on solvable lie groups linear systems on lie groups are a natural generalization of linear systems on euclidian spaces for such systems this paper studies controllability by taking into consideration the eigenvalues of an associated derivation mathcal d when the state space is a solvable connected lie group controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation mathcal d has only pure imaginary eigenvalues for bounded systems on nilpotent lie groups such conditions are also necessary

the influence of fractional diffusion in fisher kpp equations we study the fisher kpp equation where the laplacian is replaced by the generator of a feller semigroup with power decaying kernel an important example being the fractional laplacian in contrast with the case of the standard laplacian where the stable state invades the unstable one at constant speed we prove that with fractional diffusion generated for instance by a stable levy process the front position is exponential in time our results provide a mathematically rigorous justification of numerous heuristics about this model

remarks on certain criteria for detection of number of signals in this paper we derive the asymptotic distribution of the logarithm of the likelihood ratio statistic for testing the hypothesis that the number of signals is equal to q against the alternative that it is equal to k specified for a special case this distribution is not chi square the above statistic also rises in studying consistency property of the mdl criterion and aic for detection of the number of signals

reaction diffusion on random spatial networks with scale free jumping rates via effective medium theory we study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites an average transport network is found using a recently developed variant of the effective medium approximation ema that is capable of dealing with these long range connections using a feynman kac argument in the effective medium we derive an estimate on the size of the infected domain and reproduce the known result of its exponential growth in time we hereby demonstrate the applicability of long range ema to dynamical processes on networks more intricate than simple diffusion

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

front propagation in reactive systems with anomalous diffusion abstract we study front propagation of reactive fields in systems whose diffusive behavior is anomalous both superdiffusive and subdiffusive the features of the front propagation depend not only on the scaling exponent x t 2 t 2 but also on the detailed shape of the probability distribution of the diffusive process from the analysis of different systems we have three possible behavior of front propagation the usual fisher kolmogorov like scenario i e the field has a spatial exponential tail moving with constant speed v f and thickness the field has a spatial exponential tail but v f and change in time as a power law and finally the field has a spatial power law tail and v f increases exponentially in time a linear analysis of the front tail is in quantitative agreement with the numerical simulations it is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation moreover if the probability distribution of the transport process follows the scaling relation given by the flory argument the front propagation is standard fisher kolmogorov like even in presence of super or sub diffusion

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

source enumeration for high resolution array processing using improved gerschgorin radii without eigendecomposition accurate detection of sources with low complexity is of considerable interest in practical applications of high resolution array processing this paper addresses a new computationally efficient method for source enumeration by using enhanced gerschgorin radii without eigendecompositofion the proposed method can calculate the gerschgorin radii in a more efficient manner in which the additive background noise can be efficiently suppressed and the computational complexity can be considerably reduced therefore the method is more accurate and computationally attractive furthermore the method does not rely on the eigenvalues of a covariance matrix or the signal noise power making it robust against deviations from the assumption of spatially white noise model numerical results are presented to demonstrate the performance of the method

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

effects of cross diffusion on turing bifurcations in two species reaction transport systems we study the turing bifurcation in general two species reaction transport systems where particle dispersal is governed by diffusion including cross diffusion by performing a linear stability analysis we find conditions for the turing instability and compare the results with the standard turing conditions we apply our results to two model systems the lengyel epstein model and the brusselator and find strong effects of cross diffusion on the turing bifurcation

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

large dimensional analysis of robust m estimators of covariance with outliers a large dimensional characterization of robust m estimators of covariance or scatter is provided under the assumption that the dataset comprises independent essentially gaussian legitimate samples as well as arbitrary deterministic samples referred to as outliers building upon recent random matrix advances in the area of robust statistics we specifically show that the so called maronna m estimator of scatter asymptotically behaves similar to well known random matrices when the population and sample sizes grow together to infinity the introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products with a constant weight for all legitimate samples and different weights for the outliers a fine analysis of this structure reveals importantly that the propensity of the m estimator to attenuate or enhance the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples thus robust m estimators can bring substantial benefits over more simplistic estimators such as the per sample normalized version of the sample covariance matrix which is not capable of differentiating the outlying samples the analysis shows that within the class of maronna s estimators of scatter huber estimator is more favorable in a sense to be defined for rejecting outliers than classical alternatives such as tyler s scale invariant estimator often preferred in the literature in fact the analysis reveals that estimators similar to tyler s run the risk of enhancing instead of mitigating some outliers

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

multiplicity results for a non local problem with concave and convex nonlinearities abstract we shall prove a multiplicity result for a non local problem with a super critical nonlinearity of the form 1 s u u u p 2 u u q 2 in u 0 on r n where n is a bounded domain 0 s 1 and 1 q 2 p as a consequence of our results for each p 2 there exists 0 such that for each 0 problem 1 has a sequence of solutions with a negative energy this result is new for super critical values of p we shall also explore the existence of symmetric solutions for symmetric domains our methodology is based on a variational principle established by one of the authors that allows one to deal with super critical problems when the standard euler lagrange functional is restricted to certain convex sets

reaction subdiffusion equations to analyze possible generalizations of reaction diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion a ensuremath rightarrow b we derive the corresponding kinetic equations for the local a and b concentrations their form is rather unusual the parameters of the reaction influence the diffusion term in the equation for a component a a consequence of the non markovian nature of subdiffusion the equation for the product contains a term which depends on the concentration of a at all previous times our discussion shows that reaction subdiffusion equations may not resemble the corresponding reaction diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

dimension estimation in noisy pca with sure and random matrix theory principal component analysis pca is one of the best known methods for dimensionality reduction perhaps the most important problem in using pca is to determine the number of principal components pcs or equivalently choose the rank of the loading matrix many methods have been proposed to deal with this problem but almost all of them fail in the important practical case when the number of observation is comparable to the number of variables i e the realm of random matrix theory rmt in this paper we propose to use stein s unbiased risk estimator sure to estimate with some assistance from rmt the number of principal components the method is applied both on simulated and real functional magnetic resonance imaging fmri data and compared to bic and the laplace method

independent component analysis based source number estimation and its comparison for mechanical systems abstract it has been challenging to correctly separate the mixed signals into source components when the source number is not known a priori in this paper we propose a novel source number estimation based on independent component analysis ica and clustering evaluation analysis we investigate and benchmark three information based source number estimations akaike information criterion aic minimum description length mdl and improved bayesian information criterion ibic all the above methods are comparatively studied in both numerical and experimental case studies with typical mechanical signals the results demonstrate that the proposed ica based source number estimation with nonlinear dissimilarity measures performs more stable and robust than the information based ones for mechanical systems

multifractal structure of chaotically advected chemical fields the structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two dimensional flow is investigated we focus our attention on the nonuniformities of the h older exponent of the resulting filamental chemical field they appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold nonuniformities of the h older exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements we study how this affects the scaling exponents of the structure functions displaying anomalous scaling and relate the scaling exponents to the distribution of local lyapunov exponents of the advection dynamics theoretical predictions are compared with numerical experiments

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

reduced rank mdl method for source enumeration in high resolution array processing this paper proposes a reduced rank minimum description length mdl method to enumerate the incident waves impinging on a uniform linear array ula first a new observation data and a reference signal are formed from sensor data by means of the shift invariance property of the ula a cross correlation between them is calculated which is able to capture signal information and efficiently suppress additive noise second the normalized cross correlation is used as initial information for a recursion procedure to quickly partition the observation data into two orthogonal components in a signal subspace and a reduced rank noise subspace the components in the noise subspace are employed to calculate the total code length that is required to encode the observation data finally the model with the shortest code length namely the minimum description length is chosen as the best model unlike the traditional mdl methods this method partitions the observation data into the cleaner signal and noise subspace components by means of the recursion procedure avoiding the estimation of a covariance matrix and its eigendecomposition thus the method has the advantage of computational simplicity its performance is demonstrated via numerical results

chernoff information between gaussian trees abstract in this paper we deal with gaussian graphical classifiaction problem we aim to provide a systematic study of the relationship between chernoff information and topological as well as algebraic properties of the corresponding gaussian graphs for the underlying graphical testing problems we first show the relationship between chernoff information and generalized eigenvalues of the associated covariance matrices it is then proved that chernoff information between two gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees in this way we provide a sequence of equivalent gaussian tree pairs in terms of chernoff information under our proposed grafting operations bottleneck gaussian trees namely gaussian trees connected by one such operation can thus be simplified into two 3 node gaussian trees thereafter we provide a thorough study about how chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations as well as partial ordering in the end we propose an optimal linear dimension reduction method based on generalized eigenvalues for the purpose of classification which is proved to achieve maximum chernoff information among all linear transformations

detection of the number of signals a predicted eigen threshold approach a novel method for detecting the number of signals incident upon an array of sensors is described the method is based on finding upper thresholds for the observed eigenvalues of the correlation matrix of the array output the asymptotic normality of the multifold eigenvalue estimate is used to derive a prediction formula for the thresholds due to the fact that the detection error of the method can be controlled by a parameter t the performance of the method is superior to the mdl minimum description length under low snr signal to noise ratio and superior to an aic akaike information criterion under high snr the distribution functions of both missing and false alarm errors are evaluated so that the performance of the method can be analyzed and so that t can be chosen simulation results are presented to confirm the analysis

source enumeration in array processing using a two step test we investigate the problem of enumerating source signals impinging on an array of sensors given noisy limited observations the assumption of a large sample size is indispensable for most of the existing approaches so that they suffer significant performance degradation when only a small number of samples is available random matrix theory can capture the asymptotic distributions of the eigenvalues accurately and provide good approximations even for a finite sample size in this paper we take into account the relationship and distributions of the noise and signal eigenvalues based on random matrix theory and derive a simple two step test numerical simulations demonstrate that it is capable of correctly determining the number of sources in the case of small sample sizes

finite sample approximation results for principal component analysis a matrix perturbation approach principal component analysis pca is a standard tool for dimensional reduction of a set of n observations samples each with p variables in this paper using a matrix perturbation approach we study the nonasymptotic relation between the eigenvalues and eigenvectors of pca computed on a finite sample of size n and those of the limiting population pca as n oo as in machine learning we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample pca and population pca under a spiked covariance model in addition we also consider the relation between finite sample pca and the asymptotic results in the joint limit p n with p n c we present a matrix perturbation view of the phase transition phenomenon and a simple linear algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit moreover our analysis also applies for finite p n where we show that although there is no sharp phase transition as in the infinite case either as a function of noise level or as a function of sample size n the eigenvector of sample pca may exhibit a sharp loss of tracking suddenly losing its relation to the true eigenvector of the population pca matrix this occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise whose eigenvector points in a random direction

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

spectrum sensing in the presence of multiple primary users we consider multi antenna cooperative spectrum sensing in cognitive radio networks when there may be multiple primary users a detector based on the spherical test is analyzed in such a scenario based on the moments of the distributions involved simple and accurate analytical formulae for the key performance metrics of the detector are derived the false alarm and the detection probabilities as well as the detection threshold and receiver operation characteristics are available in closed form simulations are provided to verify the accuracy of the derived results and to compare with other detectors in realistic sensing scenarios

turing bifurcation in a reaction diffusion system with density dependent dispersal abstract motivated by the recent finding n kumar g m viswanathan v m kenkre physica a 388 2009 3687 that the dynamics of particles undergoing density dependent nonlinear diffusion shows sub diffusive behaviour we study the turing bifurcation in a two variable system with this kind of dispersal we perform a linear stability analysis of the uniform steady state to find the conditions for the turing bifurcation and compare it with the standard turing condition in a reaction diffusion system where dispersal is described by simple fickian diffusion while activator inhibitor kinetics are a necessary condition for the turing instability as in standard two variable systems the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator we apply these results to two model systems the brusselator and the gierer meinhardt model

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

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

detection of signals by information theoretic criteria general asymptotic performance analysis detecting the number of sources is a well known and a well investigated problem in this problem the number of sources impinging on an array of sensors is to be estimated the common approach for solving this problem is to use an information theoretic criterion like the minimum description length mdl or the akaike information criterion aic although it has been gaining much popularity and has been used in a variety of problems the performance of information theoretic criteria based estimators for the unknown number of sources has not been sufficiently studied yet in the context of array processing the performance of such estimators were analyzed only for the special case of gaussian sources where no prior knowledge of the array structure if given is used based on the theory of misspecified models this paper presents a general asymptotic analysis of the performance of any information theoretic criterion based estimator and especially of the mdl estimator in particular the performance of the mdl estimator which assumes gaussian sources and structured array when applied to gaussian sources is analyzed in addition it is shown that the performance of a certain mdl estimator is not very sensitive to the actual distribution of the source signals however appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the mdl estimator simulation results show a good fit between the empirical and the theoretical results

flexible detection criterion for source enumeration in array processing two new procedures are proposed using random matrix theory to evaluate theoretically the performance of the bayesian information criterion bic for source enumeration in array processing the proposed procedures can predict precisely the estimation results of the bic also they shed light on the behavior of the bic in order to remedy the limitation of the bic that is the bic tends to underestimate the number of sources by one in most cases an extra parameter is introduced into the bic formulation which results in a new criterion for source enumeration referred to as the flexible detection criterion fdc by carefully choosing this parameter the fdc is capable of substantially reducing the probability of underestimation which is validated by theoretical analysis and simulations note that the fdc is consistent and has low computational complexity as the bic

pattern formation of a diffusive predator prey model with strong allee effect and nonconstant death rate abstract this paper is concerned with the pattern formation of a diffusive predator prey model with strong allee effect in the prey population and nonconstant death rate for the predator the necessary and sufficient conditions of turing instability are explicitly obtained by regarding the diffusion coefficient d 2 of the predator as bifurcation parameter we demonstrate that at the critical value of the bifurcation parameter d 2 a turing bifurcation occurs i e a pattern arises the conditions for the stability of the pattern are also derived in detail our results show that strong allee effect in our model plays a crucial role in the formation of stationary pattern which is a strong contrast to the case without strong allee effect