bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

the complex cepstrum of higher order cumulants and nonminimum phase system identification a computationally efficient identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase system the method is based on the idea of computing the complex cepstrum of higher order cumulants of the system output in particular the differential cepstrum parameters of the nonminimum phase impulse response are estimated directly from higher order cumulants by least squares solution or two dimensional fft operations the method reconstructs the minimum phase and maximum phase impulse response components separately it is flexible enough to be applied on autoregressive ar moving average ma or arma system without a priori knowledge of the type of the system benchmark simulation examples demonstrate the effectiveness of the method even with short length data records

on a condition of higher order spectra factorization this paper presents a necessary and sufficient condition that guarantees that a higher order cumulant spectrum is factorizable in an appropriate product form this condition provides a way to check whether or not a given higher order spectrum is compatible with the stronger hypothesis of white generation where the analyzed signal is obtained by exciting a linear time invariant system with a white stationary non gaussian noise the condition is based on symmetries of higher order spectra and on a generalization of a formula proposed by marron et al 1990 for third order spectra phase unwrapping it is expressed at any order as an identity relating six higher order spectrum values this test can be applied directly on higher order spectral data and does not require transformations of the data or application of elaborate algorithms

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

a collection of problems for which gaussian elimination with partial pivoting is unstable a significant collection of two point boundary value problems is shown to give rise to linear systems of algebraic equations on which gaussian elimination with row partial pivoting is unstable when standard solution techniques are used

global classical solutions to the one dimensional free boundary problem for compressible non isothermal liquid crystal flow with large data abstract in this paper we study the free boundary problem of compressible non isothermal liquid crystal flow in dimension one for the vacuum interface condition that is the density is assumed to vanish and connect to vacuum continuously we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data

on the measure of the set of factorizable polynomial bispectra in a recent work 1989 the authors have shown that factorization of bispectrum is not always possible in the present work they show that the subset of factorizable bispectra has lebesgue measure zero in the set of polynomial bispectra i e those that are obtained from finite support bicumulants hence a polynomial bispectrum cannot almost always be exactly realized as that of the output of a linear model driven by a third order white input this result can be generalized to multidimensional polynomial bispectra although it follows that a linear model driven by a third order white input cannot almost always realize a given bispectrum or a bicumulant sequence the use of a linear model as an approximation in certain applications can be justified if the computed bispectrum has an index value close to unity

a free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension abstract in this paper we consider the free boundary problem for a simplified version of ericksen leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one we obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

system reconstruction from higher order spectra slices we consider the problem of system reconstruction from higher order spectra hos slices we establish that the impulse response of a complex system can be reconstructed up to a scalar and a shift based on any pair of hos slices as long as the distance between the two slices satisfies a certain condition one slice is sufficient for the reconstruction in the case of a real system we propose a cepstrum based method for system reconstruction we also propose a new method for the reconstruction of the system fourier phase based on the phase of any odd indexed bispectrum slice being able to choose the slices to be used in the reconstruction allows us to avoid bispectrum regions dominated by noise

deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes nongaussian linear processes are considered it is shown that the phase of the transfer function can be estimated under broad conditions this is not true of gaussian linear processes and in this sense gaussian linear processes are atypical the asymptotic behavior of a phase estimate is determined the phase estimates make use of bispectral estimates these ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase a number of computational illustrations are given

global classical solutions to the 1 d vacuum free boundary problem for full compressible navier stokes equations with large data the vacuum free boundary problem of one dimensional non isentropic compressible navier stokes equations with large initial data is investigated in this paper the fluid is initially assumed to occupy a finite interval and connect to the vacuum continuously at the free boundary which is often considered in the gas vacuum interface problem using the method of lagrangian particle path we derive some point wise estimates and weighted spatial and time energy estimates for the classical solutions then the global existence and uniqueness of classical solutions are shown and the expanding speed for the free boundary is proved to be finite the main difficulty of this problem is the degeneracy of the system near the free boundary previous results are only for the solutions with low regularity cf g q chen and m kratka commun partial differ equations 27 907 943 2002

global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions we consider weak solutions to a simplified ericksen leslie system of two dimensional compressible flow of nematic liquid crystals an initial boundary value problem is first studied in a bounded domain by developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity nabla mathbf d 2 mathbf d in the equations of angular momentum on the direction field and adapting the standard three level approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy then the cauchy problem with large initial data is investigated and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem provided that the second component of initial data of the direction field satisfies some geometric angle condition

identification of non minimum phase linear stochastic systems abstract the identification of time invariant non minimum phase stochastic systems driven by non gaussian white noise is considered given only the noisy observations of the system output a two step procedure is proposed in the first step a spectrally equivalent system is estimated using a standard technique that exploits only the second order statistics of the measurements in the second step a partial set of 4th order cumulants of the measurements is exploited to resolve the location of the system zeros knowledge of the probability distribution of the driving noise is not required strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are also presented in support of the theory

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

fourier series based nonminimum phase model for statistical signal processing in this paper a parametric fourier series based model fsbm for or as an approximation to an arbitrary nonminimum phase linear time invariant lti system is proposed for statistical signal processing applications where a model for lti systems is needed based on the fsbm a minimum phase linear prediction error lpe filter for amplitude estimation of the unknown lti system together with the cramer rao cr bounds is presented then an iterative algorithm for obtaining the optimum lpe filter with finite data is presented that is also an approximate maximum likelihood algorithm when data are gaussian then three iterative algorithms using higher order statistics hos with finite non gaussian data are presented to estimate parameters of the fsbm followed by some simulation results as well as some experimental results with real speech data to support the efficacy of the proposed algorithms using the fsbm finally we draw some conclusions

higher order spectrum estimation via noncausal autoregressive modeling and deconvolution a method for nonminimum phase finite impulse response fir system identification is introduced that is based on noncausal autoregressive ar modeling of the third moments of the output data the magnitude and phase response of the fir system are expressed in terms of the parameters of a noncausal ar model the set of ar parameters is obtained by solving an overdetermined linear system of equations that is formed using third moment estimates of the output data the performance of the method is illustrated with simulation examples

local well posedness of dynamics of viscous gaseous stars we establish the local in time well posedness of strong solutions to the vacuum free boundary problem of the compressible navier stokes poisson system in the spherically symmetric and isentropic motion our result captures the physical vacuum boundary behavior of the lane emden star configurations for all adiabatic exponents gamma frac 6 5

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

well posedness for compressible euler equations with physical vacuum singularity an important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states the main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary where the characteristic speeds u c coincide and have unbounded spatial derivative since c behaves like x 1 2 close to the boundary in this paper we overcome this difficulty by presenting a new formulation and new energy spaces we establish the local in time well posedness of one dimensional compressible euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity 2009 wiley periodicals inc

an iterative algorithm for signal reconstruction from bispectrum the author presents a procedure to reconstruct a minimum or nonminimum phase discrete time signal from its bispectrum the algorithm is iterative and based on using the method of projections onto convex sets pocs in an inner product space in which the vector space consists of real sequences and vector addition is defined in terms of the convolution operation prior information such as an energy bound on the signal can be incorporated into this algorithm an algorithm that employs energy information was found to produce better results in simulation than one that ignores energy constraints

on nonlinear asymptotic stability of the lane emden solutions for the viscous gaseous star problem abstract this paper proves the nonlinear asymptotic stability of the lane emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant lies in the stability range 4 3 2 it is shown that for small perturbations of a lane emden solution with same mass there exists a unique global in time strong solution to the vacuum free boundary problem of the compressible navier stokes poisson system with spherical symmetry for viscous stars and the solution captures the precise physical behavior that the sound speed is c 1 2 holder continuous across the vacuum boundary provided that lies in 4 3 2 the key is to establish the global in time regularity uniformly up to the vacuum boundary which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary density and velocity to those of the lane emden solution with detailed convergence rates and detailed large time behaviors of solutions near the vacuum boundary in particular it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the lane emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem 1 1 2a 1 1 2b 1 1 2c 1 1 2d 1 1 2e 1 1 2f are determined by the initial mass distribution and the total mass to overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals involving both lower order and higher order derivatives and space time weighted energy estimates the constructions of these weighted nonlinear functionals and space time weights depend crucially on the structures of the lane emden solution the balance of pressure and gravitation and the dissipation finally the uniform boundedness of the acceleration of the vacuum boundary is also proved

linear system solvers for boundary value odes abstract we investigate the stability properties of several linear system solvers for solving boundary value odes we consider the compactification algorithm gaussian elimination with row partial pivoting and a qr algorithm applied to linear systems arising from solving bvps for which the matrix is block bidiagonal except for bordering along the last n rows and columns we will particularly compare auto s original linear solver an lu decomposition with partial pivoting and our implementation of the analogous qr algorithm to auto two other factors the underlying continuation strategy and mesh selection strategy may affect the stability of the linear system solver for ode continuation codes as well and are also discussed in our numerical investigations

on multi dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain abstract we study the global existence of weak solutions to a multi dimensional simplified ericksen leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain r n where n 2 or 3 by exploiting a maximum principle nirenberg s interpolation inequality and a smallness condition imposed on the n th component of initial direction field d 0 to overcome the difficulties induced by the supercritical nonlinearity d 2 d in the equations of angular momentum and then adapting a modified three dimensional approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions to the initial boundary problem with large initial energy and without any smallness condition on the initial density and velocity

on statistical identification of a class of linear space invariant image blurs using non minimum phase arma models an optimal statistical parameter estimation technique was previously presented by the authors ibid vol assp 34 p 963 72 for the identification of a class of unknown linear systems that can be modeled by a zero phase finite impulse response filter this technique is now extended to a more general case using a spectrally equivalent minimum phase semp representation of the system the original zero phase system parameters are computed from the semp model parameters

arma bispectrum approach to nonminimum phase system identification an identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase fir finite impulse response system the method is based on parametric modeling of the third moments of the output sequence and uses causal and anticausal autoregressive moving average arma models the magnitude and phase response of the system are expressed in terms of the ar parameters of the arma models in particular the ar part of the causal arma model captures the minimum phase component of the system and the ar part of the anticausal arma captures the maximum phase component both sets of parameters are obtained by solving overdetermined linear systems of equations a model order selection criterion based on third order moments is proposed the arma bispectrum approach is compared to more conventional approaches for magnitude and phase reconstruction it is demonstrated that the proposed identification procedure exhibits improved modeling performance the method does not require knowledge of the non gaussian noise distribution

some numerical aspects of computing inertial manifolds various aspects of the problem of computing inertial manifolds for dissipative partial differential equations are investigated in particular methods are proposed which generalize previous ones and improve upon some of their limitations the basic ideas and techniques are exemplified mainly for the kuramoto sivashinsky equation this equation is chosen because it is fairly simple but the dynamics are sufficiently complicated and it is a classical case for which a considerable amount of computational experience has been well documented

cumulant based blind identification of linear multi input multi output systems driven by colored inputs the blind identification problem of a linear multi input multi output mimo system is widely noticed by many researchers in diverse fields due to its relevance to blind signal separation however such a problem is ill posed and has no unique solution therefore we can only find a solution of the problem within an equivalence class we address the blind identification problem of the linear mimo system driven by unobservable colored inputs using higher order statistics hos particularly the fourth order cumulants of the outputs where the unobservable inputs are mutually independent but temporally colored linear processes we first define the set which is denoted by s of stable scalar transfer functions and then define the notion of a generalized permutation matrix which is abbreviated by a g matrix over s then it is shown that the transfer function matrix of an unknown system is identified only up to post multiplication by a g matrix this result is applied to identifying fir systems for blind signal separation

higher order spectrum factorization in one and two dimensions with applications in signal modeling and nonminimum phase system identification the problem of modeling a given higher order spectrum as that of the output of a linear time invariant lti system driven by a higher order white random signal is discussed this can be posed as a higher order spectrum factorization problem a theorem is provided to discuss the existence of such a factorization a fast algorithm for efficient implementation of the higher order spectrum factorization if it exists is then proposed the results are extended to two dimensions straightforwardly the relationship of the higher order spectrum factorization problem to the well known power spectrum factorization is also discussed as applications the problems of identification of nonminimum phase lti systems given the higher order statistics of both input and output processes and phase reconstruction are presented experimental results are provided for the one dimensional case

numerical solution of constrained optimal control problems with parameters this article presents a numerical solution technique for constrained optimal control problems that contain parameters here the state control and parameter inequality constraints are accommodated via an extended penalty function this penalty function takes on large values when the constraints are violated and small values when the constraints are satisfied using the calculus of variation it is shown that the first order necessary conditions for optimality are in the form of a two point boundary value problem involving differential and algebraic equations bvp dae a multiple shooting continuation method is developed for solving this bvp dae two examples are presented to demonstrate the effectiveness of the solution approach developed in the paper

the cross bicepstrum definition properties and application for simultaneous reconstruction of three nonminimum phase signals the recovery of three signals from their cross bispectrum or the identification of the impulse responses of three parallel linear time invariant lti systems from the cross bispectrum of their system outputs by computing the complex cepstrum of the cross bispectrum as long as the signals or impulse responses have no zeros on the unit circle is discussed it is shown that the three signals can be separated completely and approximately recovered in the cross bicepstrum domain except for their magnitude and linear phase factors the computation of the cross bicepstrum can be seen as a method for the simultaneous computation of the ordinary complex cepstra of three nonminimum phase signals without the need for phase unwrapping both least squares and fast fourier transform fft based methods for computing the bicepstral coefficients are presented simulation examples of signal reconstruction in gaussian white and nongaussian colored noise and of system identification are included the results are extended to nth order cross spectra and the factorization problem for these spectra is discussed

singular value decomposition based ma order determination of non gaussian arma models singular value decomposition svd based moving average ma order determination of non gaussian processes using higher order statistics is addressed it is shown that the ma order determination of autoregressive moving average arma models is equivalent to the rank determination of a certain error matrix and a svd approach is proposed its simplified form is applied to pure ma models to improve the robustness of the order selection a combination of the svd and the product of diagonal entries pode test is proposed some interesting applications of the two svd approaches are presented and simulations verify their performance

blind convolution using signal reconstruction from partial higher order cepstral information a blind deconvolution scheme for the reconstruction of a signal that propagates in the presence of reverberation and additive noise is presented the reconstruction employs data collected by two receivers placed well apart from each other each received sequence consists of the convolution of the transmitted signal with a channel applying the bicepstrum iterative reconstruction algorithm on the differences of the cepstra coefficients of the two received sequences the cepstra coefficients of the transmission channels can be computed and used for the reconstruction of the transmitted signal the deconvolution can be performed even if the data sequences are corrupted by additive noise the computation of the cepstra coefficients is based on the cross bispectrum if the noise processes present in the observation sequences are zero mean and uncorrelated while the bicepstrum of each observation is used if the noise processes are zero mean correlated with a symmetric probability density function

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x

linear bispectrum of signals and identification of nonminimum phase fir systems driven by colored input the identification of non minimum phase finite impulse response fir systems driven by third order stationary colored signals that are not linear processes is addressed modeling the linear part of the bispectrum of a signal is discussed the bispectrum of a signal is decomposed into two multiplicative factors the linear bispectrum is defined as the factor of the bispectrum that can exactly be modeled using a third order white noise driven linear shift invariant lsi system the linear bispectrum of the output of the unknown lsi system is represented using an arma autoregressive moving average model where the ma parameters correspond to the unknown fir system impulse response coefficients and the ar parameters model the linear bispectrum of the input signal an algorithm for identifying the ma and ar parameters is given how the proposed method is different from fitting an arma model directly to the bicumulants or the bispectrum of the system output is discussed the method is applied to blur identification

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated