recursion operators and non local symmetries a new formulation of recursion operators is presented which eliminates diffi culties associated with integro differential operators this interpretation treats recursion operators and their inverses on an equal footing efficient techniques for constructing non local symmetries of differential equations result

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

on the local systems hamiltonian in the weakly non local poission brackets we study in this work the important class of non local poisson brackets pb which we call weakly non local they appeared recently in some investigations in soliton theory however there was no theory of such brackets except for very special first order case even in this case the theory was not developed enough in particular we introduce the physical forms and find casimirs momentum and canonical forms for the most important hydrodynamic type pb of that kind and their dependence on the boundary conditions

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

a fast algorithm for nonunitary joint diagonalization and its application to blind source separation a fast algorithm named complex valued fast frobenius diagonalization cvffdiag is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex valued target matrices it adopts a multiplicative update to minimize the frobenius norm formulation of the approximate joint diagonalization problem at each of multiplicative iterations a strictly diagonally dominant updated matrix is obtained this scheme ensures the invertibility of the diagonalizer the cvffdiag relaxes several constraints on the target matrices and thus has much general applications furthermore the special approximation of the cost function the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm numerical simulations are conducted to illustrate good performances of the cvffdiag

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

a generalization of joint diagonalization criteria for source separation in the field of blind source separation joint diagonalization based approaches constitute an important framework leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices jade and simultaneous third order tensor diagonalization stotd algorithms however they are often restricted to the case of cumulants of order four in this paper we extend the results leading to jade and stotd to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results this also leads us to generalize some links between contrast functions and joint diagonalization criteria on which these algorithms are based in turn one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast moreover for the two generalized jade and stotd contrasts the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression whatever the cumulant order finally some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint diagonalization of cumulant matrices

nonorthogonal joint diagonalization by combining givens and hyperbolic rotations a new algorithm for computing the nonorthogonal joint diagonalization of a set of matrices is proposed for independent component analysis and blind source separation applications this algorithm is an extension of the jacobi like algorithm first proposed in the joint approximate diagonalization of eigenmatrices jade method for orthogonal joint diagonalization the improvement consists mainly in computing a mixing matrix of determinant one and columns of equal norm instead of an orthogonal mixing matrix this target matrix is constructed iteratively by successive multiplications of not only givens rotations but also hyperbolic rotations and diagonal matrices the algorithm performance evaluated on synthetic data compares favorably with existing methods in terms of speed of convergence and complexity

on the cauchy problem for the generalized camassa holm equation in this paper we consider the cauchy problem for the generalized camassa holm equation by using the littlewood paley decomposition and nonhomogeneous besov spaces we prove that the cauchy problem for the generalized camassa holm equation is locally well posed in the besov space b p r s with 1 max 1 1 p 3 3 which generalizes the local well posedness result in 1 we also establish the ill posedness result of the generalized camassa holm equation and give a blow up criterion

frequency domain blind mimo system identification based on second and higher order statistics we present a novel frequency domain framework for the identification of a multiple input multiple output mimo system driven by white mutually independent unobservable inputs the system frequency response is obtained based on singular value decomposition svd of a matrix constructed based on the power spectrum and slices of polyspectra of the system output by appropriately selecting the polyspectra slices we can create a set of such matrices each of which could independently yield the solution of they could all be combined in a joint diagonalization scheme to yield a solution with improved statistical performance the freedom to select the polyspectra slices allows us to bypass the frequency dependent permutation ambiguity that is usually associated with frequency domain svd while at the same time allows us compute and cancel the phase ambiguity an asymptotic consistency analysis of the system magnitude response estimate is performed

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

noncircular complex ica by generalized householder reflections we develop an efficient algorithm for noncircular complex independent component analysis ica by optimizing the rows of separation matrix independently using a generalized householder reflector and newton method we apply the new algorithm to the separation of complex generalized gaussian distributed ggd sources and propose a convenient way to learn the noncircularity of the sources simulation results are reported to study the convergence behavior and confirm the superior performance of the new algorithm

iterative algorithm for joint zero diagonalization with application in blind source separation a new iterative algorithm for the nonunitary joint zero diagonalization of a set of matrices is proposed for blind source separation applications on one hand since the zero diagonalizer of the proposed algorithm is constructed iteratively by successive multiplications of an invertible matrix the singular solutions that occur in the existing nonunitary iterative algorithms are naturally avoided on the other hand compared to the algebraic method for joint zero diagonalization the proposed algorithm requires fewer matrices to be zero diagonalized to yield even better performance the extension of the algorithm to the complex and nonsquare mixing cases is also addressed numerical simulations on both synthetic data and blind source separation using time frequency distributions illustrate the performance of the algorithm and provide a comparison to the leading joint zero diagonalization schemes

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

feature discovery in non metric pairwise data pairwise proximity data given as similarity or dissimilarity matrix can violate metricity this occurs either due to noise fallible estimates or due to intrinsic non metric features such as they arise from human judgments so far the problem of non metric pairwise data has been tackled by essentially omitting the negative eigenvalues or shifting the spectrum of the associated pseudo covariance matrix for a subsequent embedding however little attention has been paid to the negative part of the spectrum itself in particular no answer was given to whether the directions associated to the negative eigenvalues would at all code variance other than noise related we show by a simple exploratory analysis that the negative eigenvalues can code for relevant structure in the data thus leading to the discovery of new features which were lost by conventional data analysis techniques the information hidden in the negative eigenvalue part of the spectrum is illustrated and discussed for three data sets namely usps handwritten digits text mining and data from cognitive psychology

an analytical constant modulus algorithm iterative constant modulus algorithms such as godard 1980 and cma have been used to blindly separate a superposition of cochannel constant modulus cm signals impinging on an antenna array these algorithms have certain deficiencies in the context of convergence to local minima and the retrieval of all individual cm signals that are present in the channel we show that the underlying constant modulus factorization problem is in fact a generalized eigenvalue problem and may be solved via a simultaneous diagonalization of a set of matrices with this new analytical approach it is possible to detect the number of cm signals present in the channel and to retrieve all of them exactly rejecting other non cm signals only a modest amount of samples is required the algorithm is robust in the presence of noise and is tested on measured data collected from an experimental set up

approximate joint singular value decomposition of an asymmetric rectangular matrix set the singular value decomposition is among the most useful and widespread tools in linear algebra often in engineering a multitude of matrices with common latent structure are available suppose we have a set of matrices for which we wish to find two orthogonal matrices and such that all products are as close as possible to rectangular diagonal form we show that the problem can be solved efficiently by iterating either power iterations followed by an orthogonalization process or givens rotations the two proposed algorithms can be seen as a generalization of approximate joint diagonalization ajd algorithms to the bilinear orthogonal forms indeed if the input matrices are symmetric and the optimization problem reduces to that of orthogonal ajd the effectiveness of the algorithms is shown with numerical simulations and the analysis of a large database of 84 electroencephalographic recordings the proposed algorithms open the road to new applications of the blind source separation framework of which we give some example for electroencephalographic data

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

blind source separation based on time frequency signal representations blind source separation consists of recovering a set of signals of which only instantaneous linear mixtures are observed thus far this problem has been solved using statistical information available on the source signals this paper introduces a new blind source separation approach exploiting the difference in the time frequency t f signatures of the sources to be separated the approach is based on the diagonalization of a combined set of spatial t f distributions in contrast to existing techniques the proposed approach allows the separation of gaussian sources with identical spectral shape but with different t f localization properties the effects of spreading the noise power while localizing the source energy in the t f domain amounts to increasing the robustness of the proposed approach with respect to noise and hence improved performance asymptotic performance analysis and numerical simulations are provided

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

a parallel dual matrix method for blind signal separation a parallel dual matrix method that considers all cases of numerical relations between a mixing matrix and a separating matrix is proposed in this letter different constrained terms are used to construct cost function for every subalgorithm these constrained terms reflect numerical relation therefore a number of undesired solutions are excluded the search region is reduced and the convergence efficiency of the algorithm is ultimately improved moreover any parallel subalgorithm is proven to converge to a desired separating matrix only if its cost function converges to zero computer simulations indicate that the algorithm efficiently performs blind signal separation

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

an improved threshold approximation for local vote decision fusion local vote decision fusion is a recently proposed method of target detection for a wireless sensor network in which individual sensors combine their decisions with those of their neighbors and report to a fusion center only if there is a majority in favor of presence the fusion center reaches a final decision about presence or absence according to whether the number of positive reports exceeds a threshold this has been shown to give a higher target detection rate for a specified false alarm rate than a system in which sensors report their initial decisions directly to the fusion center a critical aspect of the process is the appropriate setting of the threshold to achieve the specified false alarm rate we suggest here a simple alternative to the normal approximation proposed previously and demonstrate that this gives more accurate results

binary regression using an extended beta binomial distribution with discussion of correlation induced by covariate measurement errors abstract the beta binomial distribution is extended to allow negative correlations among binary variates within an experimental unit regression models are proposed for both the binary variate response rate and for the pairwise correlation between binary variates and corresponding likelihood estimation procedures are described binary regression problems are also considered in which measurement errors in the regression variables represent the sole source of overdispersion some corresponding response rate estimation procedures are described and illustrated

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

nonorthogonal approximate joint diagonalization with well conditioned diagonalizers to make the results reasonable existing joint diagonalization algorithms have imposed a variety of constraints on diagonalizers actually those constraints can be imposed uniformly by minimizing the condition number of diagonalizers motivated by this the approximate joint diagonalization problem is reviewed as a multiobjective optimization problem for the first time based on this a new algorithm for nonorthogonal joint diagonalization is developed the new algorithm yields diagonalizers which not only minimize the diagonalization error but also have as small condition numbers as possible meanwhile degenerate solutions are avoided strictly besides the new algorithm imposes few restrictions on the target set of matrices to be diagonalized which makes it widely applicable primary results on convergence are presented and we also show that for exactly jointly diagonalizable sets no local minima exist and the solutions are unique under mild conditions extensive numerical simulations illustrate the performance of the algorithm and provide comparison with other leading diagonalization methods the practical use of our algorithm is shown for blind source separation bss problems especially when ill conditioned mixing matrices are involved

the integrable kdv6 equations multiple soliton solutions and multiple singular soliton solutions abstract in this work the completely integrable sixth order nonlinear equations kdv6 are studied three distinct cases of these equations are selected for a reliable treatment multiple soliton solutions and multiple singular soliton solutions are formally derived for these equations the cole hopf transformation method combined with the hirota s bilinear sense are used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

npsa nonorthogonal principal skewness analysis principal skewness analysis psa has been introduced for feature extraction in hyperspectral imagery as a third order generalization of principal component analysis pca its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs the eigenvalues and the corresponding eigenvectors of a coskewness tensor by combining a fixed point method with an orthogonal constraint the new eigenpairs are prevented from converging to the same previously determined maxima however in general the eigenvectors of the supersymmetric tensor are not inherently orthogonal which implies that the results obtained by the search strategy used in psa may unavoidably deviate from the actual eigenpairs in this paper we propose a new nonorthogonal search strategy to solve this problem and the new algorithm is named nonorthogonal principal skewness analysis npsa the contribution of npsa lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the kronecker product of the previous eigenvector with itself instead of its orthogonal complement space we also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with psa in addition after some algebraic derivations the complexity of the presented algorithm is also greatly reduced experiments with both simulated data and real multi hyperspectral imagery demonstrate its validity in feature extraction

the cole hopf transformation and improved tanh coth method applied to new integrable system kdv6 abstract in this paper we study a new coupled system kdv6 from the point of view of constructing exact solutions for it by using the cole hopf transformation and one improved tanh coth method traveling wave solution which includes periodic and soliton solutions are formally derived

extension of integrable equations in this communication we show that there is general construction to produce non evolutionary integrable equations from a given integrable evolutionary equation to support the main theorem a few examples are explicitly given

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

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

peakon solutions of the novikov equation and properties of the data to solution map abstract we consider both the periodic and the non periodic cauchy problem for the novikov equation and discuss continuity results for the data to solution map in sobolev spaces in particular we show that the data to solution map is not globally uniformly continuous in sobolev spaces with exponent less than 3 2 to accomplish this we construct sequences of peakon solutions whose distance initially goes to zero but later becomes large

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

nonorthogonal joint diagonalization algorithm based on trigonometric parameterization the joint diagonalization technique is an important type of method for blind source separation in this paper a new approach is presented to joint diagonalization for a set of symmetric matrices with a general and not necessarily orthogonal matrix the approach performs joint diagonalization via a series of symmetric eigen decompositions including merits of simplicity effectiveness and computational efficiency simulation results demonstrate the potential improvement of the performance in the context of blind source separation

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

quadratic optimization for simultaneous matrix diagonalization simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics although objective functions in a least squares sense can be easily formulated their minimization is not trivial because constraints and fourth order terms are usually involved most known optimization algorithms are therefore subject to certain restrictions on the class of problems orthogonal transformations sets of symmetric hermitian or positive definite matrices to name a few in this paper we present a new algorithm called qdiag that splits the overall optimization problem into a sequence of simpler second order subproblems there are no restrictions imposed on the transformation matrix which may be nonorthogonal indefinite or even rectangular and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized except for one of them we apply the new method to second order blind source separation and show that the algorithm converges fast and reliably it allows for an implementation with a complexity independent of the number of matrices and therefore is particularly suitable for problems dealing with large sets of matrices

a fast algorithm for joint diagonalization with non orthogonal transformations and its application to blind source separation a new efficient algorithm is presented for joint diagonalization of several matrices the algorithm is based on the frobenius norm formulation of the joint diagonalization problem and addresses diagonalization with a general non orthogonal transformation the iterative scheme of the algorithm is based on a multiplicative update which ensures the invertibility of the diagonalizer the algorithm s efficiency stems from the special approximation of the cost function resulting in a sparse block diagonal hessian to be used in the computation of the quasi newton update step extensive numerical simulations illustrate the performance of the algorithm and provide a comparison to other leading diagonalization methods the results of such comparison demonstrate that the proposed algorithm is a viable alternative to existing state of the art joint diagonalization algorithms the practical use of our algorithm is shown for blind source separation problems

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

a comparative study of approximate joint diagonalization algorithms for blind source separation in presence of additive noise a comparative study of approximate joint diagonalization algorithms of a set of matrices is presented using a weighted least squares criterion without the orthogonality constraint an algorithm is compared with an analogous one for blind source separation bss the criterion of the present algorithm is on the separating matrix while the other is on the mixing matrix the convergence of the algorithm is proved under some mild assumptions the performances of the two algorithms are compared with usual standard algorithms using bss simulations results we show that the improvement in estimating the separating matrix resulting from the relaxation of the orthogonality restriction can be achieved in presence of additive noise when the length of observed sequences is sufficiently large and when the mixing matrix is not close to an orthogonal matrix

non orthogonal joint diagonalization in the least squares sense with application in blind source separation approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation bss algorithms a common measure of the attained diagonalization of the set is the weighted least squares wls criterion however most well known algorithms are restricted to finding an orthogonal diagonalizing matrix relying on a whitening phase for the nonorthogonal factor often such an approach implies unbalanced weighting which can result in degraded performance we propose an iterative alternating directions algorithm for minimizing the wls criterion with respect to a general not necessarily orthogonal diagonalizing matrix under some mild assumptions we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations we distinguish between hermitian and symmetrical problems using bss simulations results we demonstrate the improvement in estimating the mixing matrix resulting from the relaxation of the orthogonality restriction

conservation laws and symmetries for a nonholonomic deformed volterra equation a nonholonomic deformed volterra equation is studied its lax representation infinitely many of conservation laws and generalized commutation symmetries are given a degenerate recursion operator to generate the generalized symmetries is proposed