canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

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

signal reconstruction from the phase of the bispectrum the authors present a simple procedure the bispectrum signal reconstruction bsr algorithm to recover the fourier phase of a signal from the phase of its bispectrum by simple analogy a procedure that recovers the fourier magnitude of a signal from the magnitude of its bispectrum is also presented in addition the authors propose an iterative scheme the bicepstrum iterative reconstruction algorithm bira for the reconstruction of a finite impulse response fir sequence from only the phase of its bispectrum and they demonstrate how some a priori information on the energy of the cepstra coefficients can improve significantly the convergence rate of the algorithm both schemes are based on the key observation that the differences of the bispectrum coefficients contain all the information concerning the fourier phase of the signal whereas their sums contain the fourier magnitude information

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

uniqueness theorems on meromorphic functions sharing one value in this paper we study with a weighted sharing method the uniqueness problem of f n z k and g n z k sharing one value and obtain some results which extend the theorems given by m fang s bhoosnurmath and s dyavanal et al

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

new families of exact solutions to the integrable dispersive long wave equations in 2 1 dimensional spaces the integrable dispersive long wave equations especially the higher dimensional ones are of current interest in both physics and mathematics obtained in this paper via a symbolic computation based method are new families of exact solutions to the 2 1 dimensional integrable dispersive long wave equations sample solutions from those families are presented solitary waves are merely a special case in one family

uniqueness and value sharing of meromorphic functions in this paper we study the uniqueness of meromorphic functions and prove the following theorem let f z and g z be two non constant meromorphic functions n k two positive integers with n 3k 8 if f n z k and g n z k share 1 cm then either f z c 1e c z g z c 2e c z where c 1 c 2 and c are three constants satisfying 1 k c 1c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1 our results improves the results of fang m l fang uniqueness and value sharing of entire functions comput math appl 44 2002 823 831 7 fang and hong m l fang w hong a unicity theorem for entire functions concerning differential polynomials indian j pure appl math 32 9 2001 1343 1348 8 and lin and yi w c lin h x yi uniqueness theorems for meromorphic function indian j pure appl math 32 9 2004 121 132 9

the investigation for 2 1 dimensional eckhaus type extension of the dispersive long wave equation the 2 1 dimensional eckhaus type extension of the dispersive long wave eedlw equation is investigated which was obtained in the appropriate approximation from the basic equations of hydrodynamics though it has no painleve property we gain an auto backlund transformation abt by truncating the laurent series expansion at o w0 in particular the special one of the abt establishes a relationship between the eedlw equation and a set of three linear partial differential equations involving the well known heat equation finally many types of new exact solutions of the eedlw equation are found from the obtained abt and some proper ansatze which may be useful to explain some physical phenomena

string theory and water waves we uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non perturbatively we are able to embed the type 0a and 0b a a minimal string theories into this single framework the string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy we observe that there are several other string like limits of the system and conjecture that some of them are type iia and iib a d minimal string backgrounds we explain how these and several string like special points arise and are connected in some cases the framework endows the theories with a non perturbative definition for the first time notably we discover that the painlev iv equation plays a key role in organizing the string theory physics joining its siblings painlev i and ii whose roles have previously been identified in this minimal string context

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

the complex cepstrum and bicepstrum analytic performance evaluation in the presence of gaussian noise an analytic performance evaluation of the complex cepstrum power cepstrum bicepstrum and power bicepstrum is presented by providing approximate expressions of the bias and variance of the cepstrum parameters due to the presence of gaussian noise the approximations are based on the asymptotic properties of bispectrum estimates and they are valid for moderate signal to noise ratios snrs the assumed model consists of a deterministic signal e g multiple echoes in additive white gaussian noise and finite length data the validity of the analytic expressions is verified with monte carlo simulations performance comparisons between bicepstrum and cepstrum methods suggest that the improved performance of the bicepstrum is a function of snr a omega m and n

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

regular articlesquared eigenfunction symmetries for soliton equations part ii nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

symbolic computation and new families of exact soliton like solutions to the integrable broer kaup bk equations in 2 1 dimensional spaces in this paper new families of soliton like solutions are obtained for 2 1 dimensional integrable broer kaup equations by using the symbolic computation method developed by gao and tian sample solutions obtained from these methods are presented solitary wave solutions are merely a special case in one family the method can also be extended to other types of nonlinear evolution equations in mathematical physics

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

group theoretical analysis of dispersive long wave equations in two space dimensions abstract the symmetry algebra of an integrable dispersive long wave equation in two space dimensions is shown to be infinite dimensional and to have a kac moody virasoro structure the corresponding symmetry group is used to generate a large number of new solutions in particular solitons kinks and periodic waves these waves have wave crests of quite general shapes the results are compared to those for a nonintegrable two space dimensional dispersive long wave equation for which the symmetry group is finite dimensional and wave crests are always straight lines

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

uniqueness of meromorphic functions sharing a small function and its applications in this paper we shall study the uniqueness problems of meromorphic functions sharing a small function our results improve or generalize many previous results on value sharing of meromorphic functions

a least squares approach to blind channel identification conventional blind channel identification algorithms are based on channel outputs and knowledge of the probabilistic model of channel input in some practical applications however the input statistical model may not be known or there may not be sufficient data to obtain accurate enough estimates of certain statistics in this paper we consider the system input to be an unknown deterministic signal and study the problem of blind identification of multichannel fir systems without requiring the knowledge of the input statistical model a new blind identification algorithm based solely on the system outputs is proposed necessary and sufficient identifiability conditions in terms of the multichannel systems and the deterministic input signal are also presented

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

a transformation with symbolic computation and abundant new soliton like solutions for the 1 2 dimensional generalized burgers equation in this paper an auto backlund transformation is presented for the generalized burgers equation ut uxy uuy ux 1xuy 0 is constant by using an ansatz and symbolic computation particularly this equation is transformed into a 1 2 dimensional generalized heat equation t xy 0 by the cole hopf transformation this shows that this equation is c integrable abundant types of new soliton like solutions are obtained by virtue of the obtained transformation these solutions contain n soliton like solutions shock wave solutions and singular soliton like solutions which may be of important significance in explaining some physical phenomena the approach can also be extended to other types of nonlinear partial differential equations in mathematical physics

a novel blind deconvolution scheme for image restoration using recursive filtering we present a novel blind deconvolution technique for the restoration of linearly degraded images without explicit knowledge of either the original image or the point spread function the technique applies to situations in which the scene consists of a finite support object against a uniformly black grey or white background this occurs in certain types of astronomical imaging medical imaging and one dimensional 1 d gamma ray spectra processing among others the only information required are the nonnegativity of the true image and the support size of the original object the restoration procedure involves recursive filtering of the blurred image to minimize a convex cost function we prove convexity of the cost function establish sufficient conditions to guarantee a unique solution and examine the performance of the technique in the presence of noise the new approach is experimentally shown to be more reliable and to have faster convergence than existing nonparametric finite support blind deconvolution methods for situations in which the exact object support is unknown we propose a novel support finding algorithm

constructing families of soliton like solutions to a 2 1 dimensional breaking soliton equation using symbolic computation abstract the application of computer algebra to science has a bright future in this paper using computerized symbolic computation new families of soliton like solutions are obtained for 2 1 dimensional breaking soliton equations using an ansatz these solutions contain traveling wave solutions that are of important significance in explaining some physical phenomena the method can also be applied to other types of nonlinear evolution equations in mathematical physics

noncausal nonminimum phase arma modeling of non gaussian processes a method is presented for the estimation of the parameters of a noncausal nonminimum phase arma model for non gaussian random processes using certain higher order cepstra slices the fourier phases of two intermediate sequences h sub min n and h sub max n can be computed where h sub min n is composed of the minimum phase parts of the ar and ma models and h sub max n of the corresponding maximum phase parts under the condition that there are no zero pole cancellations in the arma model these two sequences can be estimated from their phases only and lead to the reconstruction of the ar and ma parameters within a scalar and a time shift the ar and ma orders do not have to be estimated separately but they are by product of the parameter estimation procedure through simulations it is shown that unlike existing methods the estimation procedure is fairly robust if a small order mismatch occurs since the robustness of the method in the presence of additive noise depends on the accuracy of the estimated phases of h sub min n and h sub max n the phase errors due to finite length data are studied and their statistics are derived

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

bifurcation of travelling wave solutions for the modified dispersive water wave equation abstract the method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation we obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits in different regions different types of travelling solutions including solitary wave solutions shock wave solutions and periodic wave solutions are simulated furthermore with a generalized projective riccati equation method several new explicit exact solutions are obtained

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures