extension of the adler bobenko suris classification of integrable lattice equations the classification of lattice equations that are integrable in the sense of higher dimensional consistency is extended by allowing directed edges we find two cases that are not transformable via the admissible transformations to the lattice equations in the existing classification

on preserving the robustness of an optimal control system with observers a new simple and general method is proposed in this note to preserve the robustness of the lq control system which involves state estimators the method is to minimize the estimator gain to the system input by selecting the estimator poles while the existing method is to maximize the estimator gain to the system output by introducing additional noise in the system input noise description an example demonstrates that the stability margin of the feedback system with the estimator designed by this method is even better than that of a system without the estimators

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

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

expansions over the squared solutions and difference evolution equations the completeness relation for the system of squared solutions of the discrete analog of the zakharov shabat problem is derived it allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results these include i the expansion of the potential and its variations over the system of squared solutions the expansion coefficients being the scattering data and their variations respectively thus the interpretation of the inverse scattering transform ist as a generalized fourier transform becomes obvious ii compact expressions for the trace identities through the operator for which the squared solutions are eigenfunctions iii brief exposition of the spectral theory of the operator iv direct calculation of the action angle variables based on the symplectic form of the completeness relation v the generating functional of the m operators in the lax representation vi the quantum ve

author s reply to comments on the loop transfer recovery the author disagrees with the two comments made in the above paper by rachid ibid vol 40 no 2 1995 on the new loop transfer recovery ltr technique proposed by the author ibid vol ac 32 no 9 1987 1 the technique could lead to poor results and 2 the technique cannot be applied to nonminimum phase systems referring to the first comment it is pointed out that the new technique can yield exact solutions for all plants with more than inputs or with at least one stable transmission zero the author dismisses the second as incorrect

integrable three dimensional lattices following the work of zakharov and shabat 1974 the authors derive nonlinear equations which are integrable through a discrete gel fand levitan integral equation in two resp one resp zero continuous and one resp two resp three discrete variables

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

the spectral wronskian tool and the investigation of the kdv hierarchy the recently introduced spatial transform st method for providing solutions to nonlinear evolution equations is developed when the basic equation is the spectral schrodinger equation s a fundamental tool is the spectral wronskian which allows one to take advantage of the structure of a two dimensional module for some set of solutions of s this leads easily to the kdv hierarchy contrary to the usual spectral transform or inverse scattering transform method there is no a priori assumption on the long distance behavior of the solutions a recursion operator is exhibited local conservation laws and backlund relations are also derived

lattice equations hierarchies and hamiltonian structures iii the 2d toda and kp hierarchies abstract we continue the investigation of the continuum limit properties of discrete versions of the kadomtsev petviashvili kp equation using a multiple time scale expansion in which one of the discrete variables is replaced by an infinite number of continous variables we can derive a hierarchy of integrable equations in this paper hierarchies associated with the 2d toda equation and the kp equation are treated the direct linearization of the hierarchy is obtained applying the expansion mentioned above to the free wave function in the integral equation the hamiltonian structure of the hierarchy is established via a relation between the generator of the hierarchy and an appropriate monodromy factor of the scattering problem

direct linearizing transform for three dimensional discrete integrable systems the lattice akp bkp and ckp equations a unified framework is presented for the solution structure of three dimensional discrete integrable systems including the lattice akp bkp and ckp equations this is done through the so called direct linearizing transform which establishes a general class of integral transforms between solutions as a particular application novel soliton type solutions for the lattice ckp equation are obtained

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

baklund transformations and singular integral equations abstract a systematic method for deriving backlund transformations for singular linear integral equations is presented the method leads in a natural way to the backlund transformations for the corresponding integrable nonlinear partial differential equations by repeated use of the method a hierarchy of singular integral equations and backlund transformations can be derived corresponding to so called multi modified partial differential equations specific examples include the korteweg de vries equation and the first second and third modified korteweg de vries equation the nonlinear schrodinger equation and the anisotropic heisenberg spin chain the sine gordon equation and the modified sine gordon equation

higher order svd based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems multidimensional harmonic retrieval problems are encountered in a variety of signal processing applications including radar sonar communications medical imaging and the estimation of the parameters of the dominant multipath components from mimo channel measurements r dimensional subspace based methods such as r d unitary esprit r d rare or r d music are frequently used for this task since the measurement data is multidimensional current approaches require stacking the dimensions into one highly structured matrix however in the conventional subspace estimation step e g via an svd of the latter matrix this structure is not exploited in this paper we define a measurement tensor and estimate the signal subspace through a higher order svd this allows us to exploit the structure inherent in the measurement data already in the first step of the algorithm which leads to better estimates of the signal subspace we show how the concepts of forward backward averaging and the mapping of centro hermitian matrices to real valued matrices of the same size can be extended to tensors as examples we develop the r d standard tensor esprit and the r d unitary tensor esprit algorithms however these new concepts can be applied to any multidimensional subspace based parameter estimation scheme significant improvements of the resulting parameter estimation accuracy are achieved if there is at least one of the r dimensions which possesses a number of sensors that is larger than the number of sources this can already be observed in the two dimensional case

an inverse problem for multidimensional first order systems an inverse problem associated with n first order equations in n 1 dimensions n 1 is considered given appropriate inverse data t reconstruct the potential q x0 x where q is an n n off diagonal matrix although q depends on n 1 variables it turns out that t depends on 3n 1 variables this necessitates imposing certain constraints on t i e t must be suitably characterized the characterization problem for t is solved explicitly furthermore the problem of reconstructing q is reduced to one for reconstructing a 2 2 matrix potential in two dimensions the inverse data needed for the reduced problem are obtained in closed form from t a method for solving two dimensional inverse problems has recently appeared in the literature

periodic fixed points of backlund transformations and the korteweg de vries equation a new method for studying integrable systems based on the periodic fixed points of backlund transformations bt s is presented normally the bt maps an old solution into a new solution and requires a known seed solution to get started besides this limitation it can also be difficult to qualitatively classify the result of applying the bt several times to a known solution by studying the periodic fixed points of the bt regarded as a nonlinear map in a function space integrable systems of equations of finite degree equal to the order of the fixed point and a method for the systematic classification of the solutions of the original system are obtained

deriving conservation laws for abs lattice equations from lax pairs in this paper we derive infinitely many conservation laws for abs lattice equations from their lax pairs these conservation laws can be algebraically expressed by means of some known polynomials for each equation the infinitely many conservation laws are not equivalent and are nontrivial we also show that the h1 h2 h3 q1 q2 q3 and a1 equations in the abs list share a generic discrete riccati equation

davey stewartson type equations in 4 2 and 3 1 possessing soliton solutions an integrable generalisation of a davey stewartson type system of two equations involving two scalar functions in 4 2 i e in four spatial and two temporal dimensions has been recently derived by one of the authors here we first show that there exists a reduction of this system to a single equation involving a scalar function in 4 2 we will refer to this equation as the 4 2 davey stewartson equation we then show that it is possible to reduce this equation to an equation in 3 1 which we will refer to as the 3 1 davey stewartson equation furthermore by employing the so called direct linearising method we compute 1 and 2 soliton solutions for both the 4 2 and the 3 1 davey stewartson equations

string equations for the kdv hierarchy and the grassmannian the grassmannian model for the korteweg de vries hierarchy is used to describe the translational galilean and scaling self similar solutions of this hierarchy these solutions are characterized by the string equations appearing in two dimensional quantum gravity in particular the subsets of the sato grassmannian corresponding to solutions of the string equations are found the well known adler moser rational solutions are obtained as well as a three parameter family of solutions associated with the painleve ii equation

wronskian and rational solutions of the differential difference kp equation we present the wronskian form of the n soliton solutions of the differential difference kadomtsev petviashvilli dkp equation also a wider class of rational solutions are derived using semi discrete analogue of schur polynomials and its generalization our approach is based on sato theory formalism which gives all these solutions naturally

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

mathbb z _ n graded discrete lax pairs and integrable difference equations we introduce a class of z n graded discrete lax pairs with n n matrices linear in the spectral parameter we give a classification scheme for such lax pairs and the associated discrete integrable systems we present two potential forms and completely classify the generic case many well known examples belong to our scheme for n 2 so many of our systems may be regarded as generalisations of these even at n 3 several new integrable systems arise many of our equations are mutually compatible so can be used together to form coloured lattices

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

solutions of the davey stewartson equation with non zero boundary condition the talk is linked up to the direct linearization dl of the ds ii as given in section 5 of ref 1 and uses the same notation the treatment in ref 1 is still to be supplemented by the calculation of solutions x y t x y t of the ds ii from the choice of measure and contour int limits rm d int rm d zeta ell rm ell rm sum limits rm n 1 rm n sum limits rm m 1 rm m rm c rm nm 1 over 2 rm pi i 2 oint rm d ell 1 over ell rm i rm p rm n oint rm d ell 1 over ell rm i rm q rm m 1

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

lattice equations hierarchies and hamiltonian structures ii kp type of hierarchies on 2d lattices abstract we investigate the continuum limit properties of an integrable lattice version of the kadomtsev petviashvili kp equation by applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice kp we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two dimensional lattice the direct linearization of the hierarchies is obtained applying the same limit to the free wave function in the integral equation for lattice kp on the basis of the relation between the generator of the hierarchy and the monodromy factor for the scattering problem on a line which is not involved in the continuum limit we establish the hamiltonian structure of the equations of the hierarchy

the sylvester equation and the elliptic korteweg de vries system the elliptic potential korteweg de vries lattice system is a multi component extension of the lattice potential korteweg de vries equation whose soliton solutions are associated with an elliptic cauchy kernel i e a cauchy kernel on the torus in this paper we generalize the class of solutions by allowing the spectral parameter to be a full matrix obeying a matrix version of the equation of the elliptic curve and for the cauchy matrix to be a solution of a sylvester type matrix equation subject to this matrix elliptic curve equation the construction involves solving the matrix elliptic curve equation by using toeplitz matrix techniques and analysing the solution of the sylvester equation in terms of jordan normal forms furthermore we consider the continuum limit system associated with the elliptic potential korteweg de vries system and analyse the dynamics of the soliton solutions which reveals some new features of the elliptic system in comparison to the non elliptic case

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

joint spatial division and multiplexing the large scale array regime we propose joint spatial division and multiplexing jsdm an approach to multiuser mimo downlink that exploits the structure of the correlation of the channel vectors in order to allow for a large number of antennas at the base station while requiring reduced dimensional channel state information at the transmitter csit jsdm achieves significant savings both in the downlink training and in the csit uplink feedback thus making the use of large antenna arrays at the base station potentially suitable also for frequency division duplexing fdd systems for which uplink downlink channel reciprocity cannot be exploited in the proposed scheme the multiuser mimo downlink precoder is obtained by concatenating a prebeamforming matrix which depends only on the channel second order statistics with a classical multiuser precoder based on the instantaneous knowledge of the resulting reduced dimensional effective channel matrix we prove a simple condition under which jsdm incurs no loss of optimality with respect to the full csit case for linear uniformly spaced arrays we show that such condition is approached in the large number of antennas limit for this case we use szego s asymptotic theory of toeplitz matrices to show that a dft based prebeamforming matrix is near optimal requiring only coarse information about the users angles of arrival and angular spread finally we extend these ideas to the case of a 2 d base station antenna array with 3 d beamforming including multiple beams in the elevation angle direction we provide guidelines for the prebeamforming optimization and calculate the system spectral efficiency under proportional fairness and max min fairness criteria showing extremely attractive performance our numerical results are obtained via asymptotic random matrix theory avoiding lengthy monte carlo simulations and providing accurate results for realistic finite number of antennas and users

zn graded discrete lax pairs and yang baxter maps we recently introduced a class of z n graded discrete lax pairs and studied the associated discrete integrable systems lattice equations in this paper we introduce the corresponding yang baxter maps many well known examples belong to this scheme for n 2 so for n 3 our systems may be regarded as generalizations of these in particular for each n we introduce a class of multi component yang baxter maps which include h b iii of papageorgiou et al 2010 sigma 6 003 9 p doi 10 3842 sigma 2010 033 when n 2 and that associated with the discrete modified boussinesq equation for n 3 for n 5 we introduce a new family of yang baxter maps which have no lower dimensional analogue we also present new multi component versions of the yang baxter maps f iv and f v given in the classification of adler et al 2004 commun anal geom 12 967 1007 doi 10 4310 cag 2004 v12 n5 a1

bayesian robust principal component analysis a hierarchical bayesian model is considered for decomposing a matrix into low rank and sparse components assuming the observed matrix is a superposition of the two the matrix is assumed noisy with unknown and possibly non stationary noise statistics the bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low rank and sparse outlier contributions the model is robust to a broad range of noise levels without having to change model hyperparameter settings in addition the bayesian framework allows exploitation of additional structure in the matrix for example in video applications each row or column corresponds to a video frame and we introduce a markov dependency between consecutive rows in the matrix corresponding to consecutive frames in the video the properties of this markov process are also inferred based on the observed matrix while simultaneously denoising and recovering the low rank and sparse components we compare the bayesian model to a state of the art optimization based implementation of robust pca considering several examples we demonstrate competitive performance of the proposed model

backlund transformations for integrable lattice equations we give new backlund transformations bts for some known integrable in the sense of being multidimensionally consistent quadrilateral lattice equations as opposed to the natural auto bt inherent in every such equation these bts are of two other kinds specifically it is found that some equations admit additional auto bts with backlund parameter whilst some pairs of apparently distinct equations admit a bt which connects them

linearization of the nonlinear schrodinger equation and the isotropic heisenberg spin chain abstract a new description in terms of one and the same linear inhomogeneous integral equation is proposed for the nonlinear schrodinger equation nls as well as for the equation of motion for the classical isotropic heisenberg spin chain in the continuum limit ihsc from the integral equation which contains a two fold integration over an arbitrary contour in the complex plane with an arbitrary measure one can obtain the various solutions of the nls as well as of the ihsc in a direct way without going through the details of the inverse scattering formalism well known properties such as the miura transformation the gel fand levitan equation and the lax representations for nls and ihsc can be derived as a corollary from the integral equation the treatment leads also to a few more general integrable partial differential equations which contain the nls and the ihsc as special cases

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

closed form 2 d angle estimation with rectangular arrays in element space or beamspace via unitary esprit the uca esprit is a closed form algorithm developed for use in conjunction with a uniform circular array uca that provides automatically paired source azimuth and elevation angle estimates the 2 d unitary esprit is presented as an algorithm providing the same capabilities for a uniform rectangular array ura in the final stage of the algorithm the real and imaginary parts of the ith eigenvalue of a matrix are one to one related to the respective direction cosines of the ith source relative to the two major array axes the 2 d unitary esprit offers a number of advantages over other proposed esprit based closed form 2 d angle estimation techniques first except for the final eigenvalue decomposition of a dimension equal to the number of sources it is efficiently formulated in terms of real valued computation throughout second it is amenable to efficient beamspace implementations that are presented third it is applicable to array configurations that do not exhibit identical subarrays e g two orthogonal linear arrays finally the 2 d unitary esprit easily handles sources having one member of the spatial frequency coordinate pair in common simulation results are presented verifying the efficacy of the method

subspace identification for doa estimation in massive full dimension mimo systems bad data mitigation and automatic source enumeration in this paper the direction of arrival doa estimation problem for massive multiple input multiple output mimo systems with a two dimensional 2d array also known as full dimension mimo is investigated assuming no knowledge of path number noise power path gain correlations and bad data statistics based on the variational bayesian framework a novel iterative algorithm for subspace identification operating on tensor represented data is proposed with integrated features of effective bad data mitigation and automatic source enumeration the subspace recovered from the proposed algorithm not only enables existing 2d doa estimators to be readily applied if the number of signal paths is less than the number of horizontal antennas and vertical antennas the subspaces in elevation and azimuth domains can be separately estimated from which one dimensional 1d doa estimators can be utilized thus further lowering the complexity simulation results are presented to illustrate the excellent performance of the proposed subspace recovery method and subsequent doa estimation in terms of accuracy and robustness

soliton solutions for abs lattice equations i cauchy matrix approach in recent years there have been new insights into the integrability of quadrilateral lattice equations i e partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1 1 dimensions in the scalar i e single field case there now exist classification results by adler bobenko and suris abs leading to some new examples in addition to the lattice equations of kdv type that were known since the late 1970s and early 1980s in this paper we review the construction of soliton solutions for the kdv type lattice equations and use those results to construct n soliton solutions for all lattice equations in the abs list except for the elliptic case of q4 which is left to a separate treatment

seed and soliton solutions for adler s lattice equation adler s lattice equation has acquired the status of a master equation among 2d discrete integrable systems in this paper we derive what we believe are the first explicit solutions of this equation in particular it turns out to be necessary to establish a non trivial seed solution from which soliton solutions can subsequently be constructed using the backlund transformation as a corollary we find the corresponding solutions of the krichever novikov equation which is obtained from adler s equation in a continuum limit

soliton solutions for q3 we construct n soliton solutions to the equation called q3 in the recent adler bobenko surisclassification anessentialingredientintheconstruction is the relationship of q3 0 to the equation proposed by nijhoff quispel and capel in 1983 the nqc equation this latter equation has two extra parameters and depending on their sign choices we get a 4 to 1 relationship from nqc to q3 0 this leads to a four term background solution and then to a 1 soliton solution using a b acklund transformation using the 1ss as a guide allows us to get the n soliton solution in terms of the function of the hirota miwa equation

bayesian cp factorization of incomplete tensors with automatic rank determination candecomp parafac cp tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors the existing cp algorithms require the tensor rank to be manually specified however the determination of tensor rank remains a challenging problem especially for cp rank in addition existing approaches do not take into account uncertainty information of latent factors as well as missing entries to address these issues we formulate cp factorization using a hierarchical probabilistic model and employ a fully bayesian treatment by incorporating a sparsity inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters resulting in automatic rank determination to learn the model we develop an efficient deterministic bayesian inference algorithm which scales linearly with data size our method is characterized as a tuning parameter free approach which can effectively infer underlying multilinear factors with a low rank constraint while also providing predictive distributions over missing entries extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground truth of cp rank and prevent the overfitting problem even when a large amount of entries are missing moreover the results from real world applications including image inpainting and facial image synthesis demonstrate that our method outperforms state of the art approaches for both tensor factorization and tensor completion in terms of predictive performance

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

on some integrable discrete time systems associated with the bogoyavlensky lattices abstract a new class of integrable lattice systems is introduced which are the time discretisations of the bogoyavlensky systems finite dimensional reductions of these systems are considered that give rise to integrable mappings furthermore the similarity reduction is shown to lead to higher order q difference generalisations of the discrete painleve i equation

on reductions of the discrete kadomtsev petviashvili type equations the reduction by restricting the spectral parameters k and k prime on a generic algebraic curve of degree mathcal n is performed for the discrete akp bkp and ckp equations respectively a variety of two dimensional discrete integrable systems possessing a more general solution structure arise from the reduction and in each case a unified formula for the generic positive integer mathcal n geqslant 2 is given to express the corresponding reduced integrable lattice equations the obtained extended two dimensional lattice models give rise to many important integrable partial difference equations as special degenerations some new integrable lattice models such as the discrete sawada kotera kaup kupershmidt and hirota satsuma equations in extended form are given as examples within the framework

conservation laws for nqc type difference equations this paper presents a classification of all three point conservation laws for a large class of integrable difference equations that has been described by nijhoff quispel and capel we show that every nonlinear equation from this class has at least two nontrivial conservation laws most of the conservation laws that are found are new

soliton solutions for abs lattice equations ii casoratians and bilinearization in part i soliton solutions to the abs list of multi dimensionally consistent difference equations except q4 were derived using connection between the q3 equation and the nqc equations and then by reductions in that work the central role was played by a cauchy matrix in this work we use a different approach and we derive the n soliton solutions following hirota s direct and constructive method this leads to casoratians and bilinear difference equations we give here details for the h series of equations and for q1 the results for q3 have been given earlier

on robust observer compensator design abstract this paper presents for the first time a unified explicit and restriction free set of design formulas for kalman type and luenberger type state observers and function observers with arbitrary poles in addition a new explicit and deterministic observer pole selection method for recovering the robustness of a direct state feedback system is also proposed two design examples are given in this paper these examples demonstrate the advantages of this robust recovery method the examples also show that both the luenberger type observers and the function observers generally have better stability margins this design methodology enables the design objectives of robustness order reduction and performance to be considered together in terms of observer pole selection

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

on scale mixtures of normal distributions summary the exponential power family of distributions of box tiao 1973 is shown to be a subset of the class of scale mixtures of normals the corresponding mixing distributions are explicitly obtained identifying a close relationship between the exponential power family and a further class of normal scale mixtures namely the stable distributions

the fredholm determinant method for the kdv equations abstract the fredholm determinant method is a new and rigorous way to investigate soliton equations and to construct their solutions it consists essentially of establishing a comparatively simple relation between the soliton equation and its linearized form the inverse scattering method hirota s method of constructing n soliton solutions and backlund transformations are given a new and unified interpretation moreover some new results can be obtained in this paper the fredholm determinant method is applied to the korteweg de vries kdv modified kdv and the hierarchy of generalized kdv equations

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

exact solutions of quantum mappings from the lattice kdv as multi dimensional operator difference equations the lattice kdv equation is defined on an associative algebra and a new initial value problem on the lattice kdv is considered this leads to two new families of iterative mappings discrete equations of motion which can be solved for all discrete times imposing a certain commutation relation on the algebra makes it possible to prove using a yang baxter structure that these mappings are quantum integrable maps in the classical limit the maps become a new class of integrable mappings in the sense of liouville arnold veselov lav

stokes phenomena in discrete painleve i in this study we consider the asymptotic behaviour of the first discrete painleve equation in the limit as the independent variable becomes large using an asymptotic series expansion we identify two types of solutions which are pole free within some sector of the complex plane containing the positive real axis using exponential asymptotic techniques we determine stokes phenomena effects present within these solutions and hence the regions in which the asymptotic series expression is valid from a careful analysis of the switching behaviour across stokes lines we find that the first type of solution is uniquely defined while the second type contains two free parameters and that the region of validity may be extended for appropriate choice of these parameters

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

the anisotropic heisenberg spin chain and the nonlinear schrodinger equation abstract it is shown that the solutions of the continuous anisotropic heisenberg spin chain ahsc can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the isotropic heisenberg spin chain ihsc and the nonlinear schrodinger equation nls an explicit expression is obtained for the miura transformation which maps the solutions of the ahsc on solutions of the nls in the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of painleve type as an application we discuss some new solutions of painleve iv

a finite genus solution of the h1 model the h1 model in the adler bobenko suris list i e the lattice potential kdv equation and the closely related lattice kdv equation with nijhoff s discretization are investigated a new lax pair of the h1 model is given by which integrable symplectic maps are constructed through a non linearization procedure resorting to these maps finite genus solutions of the h1 model as well as the lattice kdv equation are calculated

on the stability robustness of the continuous time lqg optimal control the robustness properties of the lqg optimal controller that is obtained by the asymptotic recovery method are discussed it is shown that in spite of the good recovered stability margins the resulting closed loop design is inherently unrobust in the sense that small parameter variations in the plant and or small realization errors in the controller may lead to an instantaneous instability of the design

on local inverse transforms the direct linearization of a nonlinear partial differential equation when it is possible is a purely algebraic process which should not imply strong conditions on the solutions at infinity thus it should be possible to manage it in such a way that the relevant inversion equation is not of the marchenko form but others e g the true gelfand levitan form which preserves the local character of the approach this is demonstrated on the korteweg de vries case

analytical performance assessment of multi dimensional matrix and tensor based esprit type algorithms in this paper we present a generic framework for the asymptotic performance analysis of subspace based parameter estimation schemes it is based on earlier results on an explicit first order expansion of the estimation error in the signal subspace obtained via an svd of the noisy observation matrix we extend these results in a number of aspects firstly we demonstrate that an explicit first order expansion of the higher order svd hosvd based subspace estimate can be derived secondly we show how to obtain explicit first order expansions of the estimation error of arbitrary esprit type algorithms and provide the expressions for r d standard esprit r d unitary esprit r d standard tensor esprit as well as r d unitary tensor esprit thirdly we derive closed form expressions for the mean square error mse and show that they only depend on the second order moments of the noise hence to apply this framework we only need the noise to be zero mean and possess finite second order moments additional assumptions such as gaussianity or circular symmetry are not needed

direct linearization approach to discrete integrable systems associated with ℤ graded lax pairs fordy and xenitidis j phys a math theor 50 2017 165205 doi 10 1088 1751 8121 aa639a recently proposed a large class of discrete integrable systems which include a number of novel integr

iteration of some discretizations of the nonlinear schrodinger equation we consider several discretizations of the nonlinear schrodinger equation which lead naturally to the study of some symmetric difference equations of the form n 1 n 1 f n we find a variety of interesting and exotic behaviour from simple closed orbits to intricate patterns of orbits and loops in the n 1 n phase plane some analytical results for a special case are also presented

boussinesq like multi component lattice equations and multi dimensional consistency various classes of one component lattice equations defined by a multi linear relation between values at the vertices of an elementary square have recently been classified using the requirement of multi dimensional consistency consistency around the cube cac here we consider multi component equations with some equations defined on the edges of the consistency cube and others on the faces of the cube some examples of this type are already known including the lattice modified boussinesq equation lmbsq we classify the edge equations into three canonical forms and derive the consequences of their cac property this restricts the form of the face equation sufficiently so that its cac property can be analyzed as a result we obtain a number of integrable multi component lattice equations some generalizing lmbsq