anderson transition in a three dimensional kicked rotor we investigate anderson localization in a three dimensional 3d kicked rotor by a finite size scaling analysis we identify a mobility edge for a certain value of the kicking strength k k c for k k c dynamical localization does not occur all eigenstates are delocalized and the spectral correlations are well described by wigner dyson statistics this can be understood by mapping the kicked rotor problem onto a 3d anderson model am where a band of metallic states exists for sufficiently weak disorder around the critical region k approximately k c we carry out a detailed study of the level statistics and quantum diffusion in agreement with the predictions of the one parameter scaling theory opt and with previous numerical simulations the number variance is linear level repulsion is still observed and quantum diffusion is anomalous with proportional t 2 3 we note that in the 3d kicked rotor the dynamics is not random but deterministic in order to estimate the differences between these two situations we have studied a 3d kicked rotor in which the kinetic term of the associated evolution matrix is random a detailed numerical comparison shows that the differences between the two cases are relatively small however in the deterministic case only a small set of irrational periods was used a qualitative analysis of a much larger set suggests that deviations between the random and the deterministic kicked rotor can be important for certain choices of periods heuristically it is expected that localization effects will be weaker in a nonrandom potential since destructive interference will be less effective to arrest quantum diffusion however we have found that certain choices of irrational periods enhance anderson localization effects

estimates on green s functions localization and the quantum kicked rotor model and phase space l2 1r it has been conjectured by a number of authors cf fgp bel that for typical parameter values for a b the wave function t satisfying equations 0 1 0 2 will be almost periodic in time hence f t the fourier transform remains localized we will prove this here assuming the parameter s in 0 2 small thus we have theorem for k stnall and a b o tside a set of small rneasure o for k 0 the following holds let 4 w t x solve 0 1 0 2 and let to 4 0 x be a sujgciently smooth function on t then is an alrnost periodic function of timbe say as an hl t valued map in particular supt ll t llhl

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

transition from localized to extended eigenstates in the ensemble of power law random banded matrices we study statistical properties of the ensemble of large n times n random matrices whose entries h ij decrease in a power law fashion h ij sim i j alpha mapping the problem onto a nonlinear sigma model with non local interaction we find a transition from localized to extended states at alpha 1 at this critical value of alpha the system exhibits multifractality and spectral statistics intermediate between the wigner dyson and poisson one these features are reminiscent of those typical for the mobility edge of disordered conductors we find a continuous set of critical theories at alpha 1 parametrized by the value of the coupling constant of the sigma model at alpha 1 all states are expected to be localized with integrable power law tails at the same time for 1 alpha 3 2 the wave packet spreading at short time scale is superdiffusive langle r rangle sim t frac 1 2 alpha 1 which leads to a modification of the altshuler shklovskii behavior of the spectral correlation function at 1 2 alpha 1 the statistical properties of eigenstates are similar to those in a metallic sample in d alpha 1 2 1 dimensions finally the region alpha 1 2 is equivalent to the corresponding gaussian ensemble of random matrices alpha 0 the theoretical predictions are compared with results of numerical simulations

a sufficient condition for gpn stability for delay differential equations in this paper the author considers a linear test delay differential equation with non constant coefficients related to the definitions of pn and gpn stability for numerical methods he defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for gpn stability

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

fast maximum likelihood for blind identification of multiple fir channels this paper develops a fast maximum likelihood method for estimating the impulse responses of multiple fir channels driven by an arbitrary unknown input the resulting method consists of two iterative steps where each step minimizes a quadratic function the two step maximum likelihood tsml method is shown to be high snr efficient i e attaining the cramer rao lower bound crb at high snr the tsml method exploits a novel orthogonal complement matrix of the generalized sylvester matrix simulations show that the tsml method significantly outperforms the cross relation cr method and the subspace ss method and attains the crb over a wide range of snr this paper also studies a fisher information fi matrix to reveal the identifiability of the m channel system a strong connection between the fi based identifiability and the cr based identifiability is established

energy growth in schrodinger s equation with markovian forcing schrodinger s equation is considered on a one dimensional torus with time dependent potential v t v x t where v is an even trigonometric polynomial and x t is a stationary markov process it is shown that when the coupling constant is sufficiently small the average kinetic energy grows as the square root of time more generally the h s norm of the wave function is shown to behave as t s 4

independent vector analysis identification conditions and performance bounds recently an extension of independent component analysis ica from one to multiple datasets termed independent vector analysis iva has been a subject of significant research interest iva has also been shown to be a generalization of hotelling s canonical correlation analysis in this paper we provide the identification conditions for a general iva formulation which accounts for linear nonlinear and sample to sample dependencies the identification conditions are a generalization of previous results for ica and for iva when samples are independently and identically distributed furthermore a principal aim of iva is identification of dependent sources between datasets thus we provide additional conditions for when the arbitrary ordering of the estimated sources can be common across datasets performance bounds in terms of the cramer rao lower bound are also provided for demixing matrices and interference to source ratio the performance of two iva algorithms are compared to the theoretical bounds

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

second order multidimensional ica performance analysis independent component analysis ica and blind source separation bss deal with extracting a number of mutually independent elements from a set of observed linear mixtures motivated by various applications this paper considers a more general and more flexible model the sources can be partitioned into groups exhibiting dependence within a given group but independence between two different groups we argue that this is tantamount to considering multidimensional components as opposed to the standard ica case which is restricted to one dimensional components the core of the paper is devoted to the statistical analysis of the blind separation of multidimensional components based on second order statistics in a piecewise stationary model we develop the likelihood and the associated estimating equations for the gaussian case we obtain closed form expressions for the fisher information matrix and the cramer rao bound of the de mixing parameters as well as the mean square error mse of the component estimates the derived mse is valid also for non gaussian data our analysis is verified through numerical experiments and its performance is compared to classical ica in various dependence scenarios quantifying the gain in the accuracy of component recovery in presence of multidimensional components

particle propagation in a random and quasi periodic potential abstract we numerically investigate the anderson transition in an effective dimension d 3 d 11 for one particle propagation in a model random and quasi periodic potential the found critical exponents are different from the standard scaling picture we discuss possible reasons for this difference

blind separation of multi dimensional components via subspace decomposition performance analysis a prevalent approach to blind separation of multi dimensional data is a two step procedure in the first step the data are assigned a one dimensional model a separating algorithm is applied according to this model this step corresponds to classical blind source separation bss in the second step the output is assigned into groups representing the multi dimensional components in this paper we consider an even more general case in which the subpartition of the components in the first step may be into elements of any dimension not necessarily one we consider a piecewise stationary model and assume that the number and dimensions of the underlying multi dimensional components are known we obtain a closed form analytical expression for the mean square error mse in the estimation of the multi dimensional components using this two step procedure as expected this approach is suboptimal in the presence of finite data size errors therefore we can predict the expected gain from using the correct model of the components over any finer decomposition thereof in addition we demonstrate the dependence of this gain on the model parameters

joint blind source separation by multiset canonical correlation analysis in this paper we introduce a simple and effective scheme to achieve joint blind source separation bss of multiple datasets using multiset canonical correlation analysis m cca j r kettenring canonical analysis of several sets of variables biometrika vol 58 pp 433 451 1971 we first propose a generative model of joint bss based on the correlation of latent sources within and between datasets we specify source separability conditions and show that when the conditions are satisfied the group of corresponding sources from each dataset can be jointly extracted by m cca through maximization of correlation among the extracted sources we compare source separation performance of the m cca scheme with other joint bss methods and demonstrate the superior performance of the m cca scheme in achieving joint bss for a large number of datasets group of corresponding sources with heterogeneous correlation values and complex valued sources with circular and non circular distributions we apply m cca to analysis of functional magnetic resonance imaging fmri data from multiple subjects and show its utility in estimating meaningful brain activations from a visuomotor task

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

on reducibility of schrodinger equations with quasiperiodic in time potentials we prove that a linear d dimensional schrodinger equation with an x periodic and t quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector the reduction is made by means of a non autonomous linear transformation of the space of x periodic functions this transformation is a quasiperiodic function of t

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

joint independent subspace analysis uniqueness and identifiability this paper deals with the identifiability of joint independent subspace analysis jisa jisa is a recently proposed framework that subsumes independent vector analysis iva and independent subspace analysis isa each underlying mixture can be regarded as a dataset therefore jisa can be used for data fusion in this paper we assume that each dataset is an overdetermined mixture of several multivariate gaussian processes each of which has independent and identically distributed samples this setup is not identifiable when each mixture is considered individually given these assumptions jisa can be restated as coupled block diagonalization cbd of its correlation matrices hence jisa identifiability is tantamount to cbd uniqueness in this work we provide necessary and sufficient conditions for uniqueness and identifiability of jisa and cbd our analysis is based on characterizing all the cases in which the fisher information matrix is singular we prove that non identifiability may occur only due to pairs of underlying random processes with the same dimension our results provide further evidence that irreducibility has a central role in the uniqueness analysis of block based decompositions our contribution extends previous results on the uniqueness and identifiability of isa iva coupled matrix and tensor decompositions we provide examples to illustrate our results

numerical solution of stochastic differential algebraic equations with applications to transient noise simulation of microelectronic circuits the transient simulation of noise in electronic circuits leads to differential algebraic equations additively disturbed by white noise for these systems we present a mathematical model based on the theory of stochastic differential equations along with an implicit two step method for their numerical treatment this numerical scheme works directly on the given structure of the equations which makes very efficient implementations possible the order of convergence is preserved the theoretical results are verified by numerical noise simulations of benchmark circuits

double coupled canonical polyadic decomposition for joint blind source separation joint blind source separation j bss is an emerging data driven technique for multi set data fusion in this paper j bss is addressed from a tensorial perspective we show how by using second order multi set statistics in j bss a specific double coupled canonical polyadic decomposition dc cpd problem can be formulated we propose an algebraic dc cpd algorithm based on a coupled rank 1 detection mapping this algorithm converts a possibly underdetermined dc cpd to a set of overdetermined cpds the latter can be solved algebraically via a generalized eigenvalue decomposition based scheme therefore this algorithm is deterministic and returns the exact solution in the noiseless case in the noisy case it can be used to effectively initialize optimization based dc cpd algorithms in addition we obtain the deterministic and generic uniqueness conditions for dc cpd which are shown to be more relaxed than their cpd counterpart we also introduce optimization based dc cpd methods including alternating least squares and structured data fusion based methods experiment results are given to illustrate the superiority of dc cpd over standard cpd based bss methods and several existing j bss methods with regards to uniqueness and accuracy

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

convergence and stability of euler method for impulsive stochastic delay differential equations this paper deals with the mean square convergence and mean square exponential stability of an euler scheme for a linear impulsive stochastic delay differential equation isdde first a method is presented to take the grid points of the numerical scheme based on this method a fixed stepsize numerical scheme is provided based on the method of fixed stepsize grid points an euler method is given the convergence of the euler method is considered and it is shown the euler scheme is of mean square convergence with order 1 2 then the mean square exponential stability is studied using lyapunov like techniques the sufficient conditions to guarantee the mean square exponential stability are obtained the result shows that the mean square exponential stability may be reproduced by the euler scheme for linear isddes under the restriction on the stepsize at last examples are given to illustrate our results

block matrix generalizations of schur s basic theorems on hadamard products abstract in a classic 1911 paper i schur gave several useful bounds for the spectral norm and eigenvalues of the hadamard entrywise product of two matrices motivated by applications to the theory of monotone and convex matrix functions we are led to consider hadamard products in which both factors are conformally partitioned block matrices and the entries of one factor are constant within each block such products are special cases of a block kronecker tensor product and it is in this context that we present generalizations of schur s results

bounded sobolev norms for linear schrodinger equations under resonant perturbations abstract we prove that the sobolev norms for a 1 d periodic schrodinger equation remain bounded under small resonant perturbations

a note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay insfdes in short in which the initial value belongs to the phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under some caratheodory type conditions on the coefficients by means of the successive approximation especially we extend the results appeared in ren et al y ren s lu n xia remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay j comput appl math 220 2008 364 372 ren and xia y ren n xia existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay appl math comput 210 2009 72 79 and zhou and xue s zhou m xue the existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay math appl 21 2008 75 83

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

new class of random matrix ensembles with multifractal eigenvectors department of physics university of florida gainesville florida 32611 february 1 2008 three recently suggested random matrix ensembles rme are linked together to represent a class of rme with multifrac tal eigenfunction statistics the generic form of the two levelcorrelation function for the case of weak and extremely strongmultifractality is suggested pacs numbers 71 25 s 72 15 rn 05 45 b

generating a fractal butterfly floquet spectrum in a class of driven su 2 systems eigenstate statistics the floquet spectra of a class of driven su 2 systems have been shown to display butterfly patterns with multifractal properties the implication of such critical spectral behavior for the floquet eigenstate statistics is studied in this work following the methodologies for understanding the fractal behavior of energy eigenstates of time independent systems on the anderson transition point we analyze the distribution profile the mean value and the variance of the logarithm of the inverse participation ratio of the floquet eigenstates associated with multifractal floquet spectra the results show that the floquet eigenstates also display fractal behavior but with features markedly different from those in time independent anderson transition models this motivated us to propose random unitary matrix ensemble called power law random banded unitary matrix ensemble to illuminate the floquet eigenstate statistics of critical driven systems the results based on the proposed random matrix model are consistent with those obtained from our dynamical examples with or without time reversal symmetry

level spacings at the metal insulator transition in the anderson hamiltonians and multifractal random matrix ensembles we consider orthogonal unitary and symplectic ensembles of random matrices with 1 a mathrm ln x 2 potentials which obey spectral statistics different from the wigner dyson and are argued to have multifractal eigenstates if the coefficient a is small spectral correlations in the bulk are universally governed by a translationally invariant one parameter generalization of the sine kernel we provide analytic expressions for the level spacing distribution functions of this kernel which are hybrids of the wigner dyson and poisson distributions by tuning the single parameter our results can be excellently fitted to the numerical data for three symmetry classes of the three dimensional anderson hamiltonians at the metal insulator transition previously measured by several groups using exact diagonalization

convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay in this paper we study the convergence and stability of the stochastic theta method stm for a class of index 1 stochastic delay differential algebraic equations first in the case of constrained mesh i e the stepsize is a submultiple of the delay it is proved that the method is strongly consistent and convergent with order 1 2 in the mean square sense then the result is further extended to the case of non constrained mesh where we employ linear interpolation to approximate the delay argument later under a sufficient condition for mean square stability of the analytical solution it is proved that when the stepsizes are sufficiently small the stm approximations reproduce the stability of the analytical solution finally some numerical experiments are presented to illustrate the theoretical findings

stochastic differential algebraic equations of index 1 and applications in circuit simulation we discuss differential algebraic equations driven by gaussian white noise which are assumed to have noise free constraints and to be uniformly of dae index 1 we first provide a rigorous mathematical foundation of the existence and uniqueness of strong solutions our theory is based upon the theory of stochastic differential equations sdes and the theory of differential algebraic equations daes to each of which our problem reduces on making appropriate simplifications we then consider discretization methods implicit methods are necessary because of the differential algebraic structure and we consider adaptations of such methods used for sdes the consequences of an inexact solution of the implicit equations roundoff and truncation errors are analysed by means of the mean square numerical stability of general drift implicit discretization schemes for sdes we prove that the convergence properties of our drift implicit euler scheme split step backward euler scheme trapezoidal scheme and drift implicit milstein scheme carry over from the corresponding properties of these methods applied to sdes finally we show how the theory applies to the transient noise simulation of electronic circuits

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

growth of sobolev norms in linear schrodinger equations with quasi periodic potential in this paper we consider the following problem let iu t u v x t u 0 be a linear schrodinger equation periodic boundary conditions where v is a real bounded real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector denote s t the corresponding flow map thus s t preserves the l 2 norm and our aim is to study its behaviour on h s t d s 0 our main result is the growth in time is at most logarithmic thus if h s then more precisely is proven in 1d and 2d when v is small we also exhibit examples showing that a growth of higher sobolev norms may occur in this context and is thus essentially best possible

stability criteria for differential algebraic equations with multiple delays and their numerical solutions abstract this paper is concerned with the asymptotic stability of differential algebraic equations with multiple delays and their numerical solutions first we give a sufficient condition for delay independent stability after characterizing the coefficient matrices that satisfy this stability condition we propose some practical checkable criteria for asymptotic stability then we investigate the stability of numerical solutions obtained by methods and bdf methods finally solvability and stability of a class of weakly regular delay differential algebraic equations are analyzed

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

differential algebraic equations of index 1 may have an arbitrarily high structural index it is shown that the structural index of a linear differential algebraic equation dae with constant coefficients and with index 1 may be arbitrarily high contrary to a previous result from the literature this demonstrates that pantelides algorithm applied to daes of index 1 may perform an arbitrarily high number of iterations and differentiations