the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

lax pair and first integrals of the traveling wave reduction for the kdv hierarchy abstract the traveling wave reduction of the korteweg de vries hierarchy is considered the linear system of equations associated with this hierarchy is found the lax pair is used to obtain the first integrals of the traveling wave reduction for the korteweg de vries hierarchy exact formulas for the first integrals of the hierarchy are given the first three members of the hierarchy are considered in more detail these first integrals are the examples of the integrable nonlinear differential equations with the painleve property exact solutions in the form of the soliton for the traveling wave reduction of the korteweg de vries hierarchy and its first integrals are presented

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

non isospectral scattering problems and truncation for hierarchies burgers and dispersive water waves abstract non isospectral scattering problems have been proven useful for several reasons amongst them the information that they provide about painleve truncation for entire hierarchies of integrable partial differential equations pdes we show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 1 dimensions namely burgers hierarchy and the dispersive water wave hierarchy burgers equation is well known as a model for turbulence and the dispersive water wave equations as a system governing shallow water waves we also see how these results are easily extended to related hierarchies in 2 1 dimensions our results for the 2 1 dimensional burgers hierarchy and for the 1 1 and 2 1 dimensional dispersive water wave hierarchies are all new

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

a note on the generalized symmetries of lattice hierarchies we show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential delay equations all members of these hierarchies of lattice and differential delay equations commute it is thus seen that differential delay hierarchies provide a broader context within which to place lattice hierarchies

products of three triangular matrices over commutative rings abstract we characterize commutative rings with 1 over which any n n matrix a can be written as lum where l m are lower triangular u is upper triangular and l and u have all their diagonal entries 1

graph dual regularization non negative matrix factorization for co clustering low rank matrix factorization is one of the most useful tools in scientific computing data mining and computer vision among of its techniques non negative matrix factorization nmf has received considerable attention due to producing a parts based representation of the data recent research has shown that not only the observed data are found to lie on a nonlinear low dimensional manifold namely data manifold but also the features lie on a manifold namely feature manifold in this paper we propose a novel algorithm called graph dual regularization non negative matrix factorization dnmf which simultaneously considers the geometric structures of both the data manifold and the feature manifold we also present a graph dual regularization non negative matrix tri factorization algorithm dnmtf as an extension of dnmf moreover we develop two iterative updating optimization schemes for dnmf and dnmtf respectively and provide the convergence proofs of our two optimization schemes experimental results on uci benchmark data sets several image data sets and a radar hrrp data set demonstrate the effectiveness of both dnmf and dnmtf

fourth order analogies to the painleve equations using the compatibility condition for the painleve equations several new fourth order ordinary differential equations odes that are analogies of the painleve equations are found the isomonodromic linear problems for these equations are given special solutions of the fourth order odes found are discussed the painleve test is applied to investigate several fourth order odes

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

new set of symmetries of the integrable equations lie algebra and non isospectral evolution equations ii akns system the authors define directly a new set of symmetries for the akns system and prove that they constitute an infinite dimensional lie algebra with the old symmetries they also point out the relation between the new symmetry and the non isospectral problem they use the reduction technique and point out that the nls mkdv sg and sinh g hierarchies have two sets of symmetries which constitute an infinite dimensional lie algebra these results are extensions of those of chen et al 1982

factorization and painleve analysis of a class of nonlinear third order partial differential equations a class of fully nonlinear third order partial differential equations pdes is considered this class contains several examples which have recently appeared in the literature and for which rather unusual travelling wave solutions have been given these solutions consist essentially of sums of exponentials we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling wave reduction in addition we consider the painleve analysis of this set of equations both for the original pde and also for reductions to ordinary differential equations odes no equation in the class considered survives the combination of pde and ode tests also an equation in the class considered which is known to be integrable is shown to possess only the weak painleve property our analysis therefore confirms the limitations of the painleve test as a test for complete integrability when applied to fully nonlinear pdes

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

sparsity preserving projections with applications to face recognition dimensionality reduction methods drs have commonly been used as a principled way to understand the high dimensional data such as face images in this paper we propose a new unsupervised dr method called sparsity preserving projections spp unlike many existing techniques such as local preserving projection lpp and neighborhood preserving embedding npe where local neighborhood information is preserved during the dr procedure spp aims to preserve the sparse reconstructive relationship of the data which is achieved by minimizing a l1 regularization related objective function the obtained projections are invariant to rotations rescalings and translations of the data and more importantly they contain natural discriminating information even if no class labels are provided moreover spp chooses its neighborhood automatically and hence can be more conveniently used in practice compared to lpp and npe the feasibility and effectiveness of the proposed method is verified on three popular face databases yale ar and extended yale b with promising results

bilinear factor matrix norm minimization for robust pca algorithms and applications the heavy tailed distributions of corrupted outliers and singular values of all channels in low level vision have proven effective priors for many applications such as background modeling photometric stereo and image alignment and they can be well modeled by a hyper laplacian however the use of such distributions generally leads to challenging non convex non smooth and non lipschitz problems and makes existing algorithms very slow for large scale applications together with the analytic solutions to ell p norm minimization with two specific values of p i e p 1 2 and p 2 3 we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis we first define the double nuclear norm and frobenius nuclear hybrid norm penalties and then prove that they are in essence the schatten 1 2 and 2 3 quasi norms respectively which lead to much more tractable and scalable lipschitz optimization problems our experimental analysis shows that both our methods yield more accurate solutions than original schatten quasi norm minimization even when the number of observations is very limited finally we apply our penalties to various low level vision problems e g text removal moving object detection image alignment and inpainting and show that our methods usually outperform the state of the art methods

non truncated formal series symmetries and generalized virasoro algebra akns type breaking soliton equation using the formal series symmetry approach we obtain three sets of generalized symmetries of the akns type breaking soliton equation these sets of formal series symmetries are not truncated except that the arbitrary functions of the formal series symmetries are fixed as polynomials of the corresponding independent variables differing from the truncated cases such as kp and toda field equations these non truncated symmetries constitute another type of generalization of the virasoro algebra

a nonisospectral extension of the volterra hierarchy to 2 1 dimensions we give a new nonisospectral generalization of the volterra lattice equation to 2 1 dimensions we use this to construct a new nonisospectral lattice hierarchy in 2 1 dimensions along with its underlying linear problem reductions yield a variety of new integrable hierarchies including generalizations of known discrete painleve hierarchies all along with their corresponding linear problems this represents an extension of previously developed techniques to the discrete case

the lower central series of subgroups of the vershik kerov group abstract we describe the derived and the lower central series of certain subgroups of the vershik kerov group we are concerned with the subgroups consisting of infinite matrices having finite number of nonzero entries in each row we consider the group of matrices over rings which are associative commutative of stable rank at most one and such that the identity can be written as a sum of two units for this case we give a complete description of the derived and the lower central series moreover we prove that every element of discussed commutator subgroups can be written as a product of a finite number of commutators

nonisospectral scattering problems and similarity reductions abstract we give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems special cases of these changes of variables involve similarity variables and similarity solutions of a hierarchy are seen to correspond to time independent solutions of an equivalent hierarchy we thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same painleve hierarchies as examples we consider the korteweg de vries hierarchy and the dispersive water wave hierarchy

generalized higher order orthogonal iteration for tensor learning and decomposition low rank tensor completion lrtc has successfully been applied to a wide range of real world problems despite the broad successful applications existing lrtc methods may become very slow or even not applicable for large scale problems to address this issue a novel core tensor trace norm minimization ctnm method is proposed for simultaneous tensor learning and decomposition and has a much lower computational complexity in our solution first the equivalence relation of trace norm of a low rank tensor and its core tensor is induced second the trace norm of the core tensor is used to replace that of the whole tensor which leads to two much smaller scale matrix tnm problems finally an efficient alternating direction augmented lagrangian method is developed to solve our problems our ctnm formulation needs only o r n nri log sqrt i n observations to reliably recover an n th order i times i times cdots times i tensor of n rank r r ldots r compared with o ri n 1 observations required by those tensor tnm methods i gg r geq r extensive experimental results show that ctnm is usually more accurate than them and is orders of magnitude faster

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

the 2 1 dimensional nonisospectral relativistic toda hierarchy related to the generalized discrete painleve hierarchy in this paper we will concentrate on the topic of integrable discrete hierarchies in 2 1 dimensions and their connection with discrete painleve hierarchies by considering a 2 1 dimensional nonisospectral discrete linear problem two new 2 1 dimensional nonisospectral integrable lattice hierarchies the 2 1 nonisospectral relativistic toda lattice hierarchy and the 2 1 nonisospectral negative relativistic toda lattice hierarchy are constructed it is shown that the reductions of the two new 2 1 nonisospectral lattice hierarchies lead to the 2 1 dimensional nonisospectral volterra lattice hierarchy and the 2 1 dimensional nonisospectral negative volterra lattice hierarchy we also obtain two new 1 1 dimensional nonisospectral integrable lattice hierarchies and two new ordinary difference hierarchies which are direct reductions of the two 2 1 nonisospectral integrable lattice hierarchies one of the two difference hierarchies yields our previously obtained generalized discrete first painleve dpi hierarchy and another one yields a generalized alternative discrete second painleve alt dpii hierarchy

new integrable equations of fourth order and higher degree related to cosgrove s equation we give a general formulation of the algorithm of fokas and ablowitz which then allows us to obtain transformations for nth order ordinary differential equations to equations of the same order but perhaps of higher degree previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by painleve and co workers either to other equations in the painleve classification or to equations of second order and second degree as an example of our approach we consider a new fourth order ordinary differential equation due to cosgrove which is believed to define a new transcendent we obtain transformations relating this equation to other fourth order ordinary differential equations of degrees 2 all of these transformations as well as the corresponding higher degree differential equations all of which have the painleve property are new

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

a fast and accurate matrix completion method based on qr decomposition and l_ 2 1 norm minimization low rank matrix completion aims to recover matrices with missing entries and has attracted considerable attention from machine learning researchers most of the existing methods such as weighted nuclear norm minimization based methods and qatar riyal qr decomposition based methods cannot provide both convergence accuracy and convergence speed to investigate a fast and accurate completion method an iterative qr decomposition based method is proposed for computing an approximate singular value decomposition this method can compute the largest r r 0 singular values of a matrix by iterative qr decomposition then under the framework of matrix trifactorization a method for computing an approximate svd based on qr decomposition csvd qr based l 2 1 norm minimization method lnm qr is proposed for fast matrix completion theoretical analysis shows that this qr decomposition based method can obtain the same optimal solution as a nuclear norm minimization method i e the l 2 1 norm of a submatrix can converge to its nuclear norm consequently an lnm qr based iteratively reweighted l 2 1 norm minimization method irlnm qr is proposed to improve the accuracy of lnm qr theoretical analysis shows that irlnm qr is as accurate as an iteratively reweighted nuclear norm minimization method which is much more accurate than the traditional qr decomposition based matrix completion methods experimental results obtained on both synthetic and real world visual data sets show that our methods are much faster and more accurate than the state of the art methods

symmetries of a 2 1 dimensional breaking soliton equation infinitely many symmetries for a 2 1 dimensional breaking soliton equation are constructed via the infinitesimal version of the dressing method these symmetries are proved to constitute an infinite dimensional lie algebra which contains some abelian and virasoro subalgebras the hierarchies of equations generated by these symmetries are also considered these hierarchies of equations are proved to be associated with the isospectral and non isospectral deformations of the akns spectral problem

the prelle singer method and painleve hierarchies we consider systems of ordinary differential equations odes of the form bk 0 where b is a hamiltonian operator of a completely integrable partial differential equation hierarchy and k k l t such systems while of quite low order and linear in the components of k may represent higher order nonlinear systems if we make a choice of k in terms of the coefficient functions of b indeed our original motivation for the study of such systems was their appearance in the study of painleve hierarchies where the question of the reduction of order is of great importance however here we do not consider such particular cases instead we study such systems for arbitrary k where they may represent both integrable and nonintegrable systems of ordinary differential equations we consider the application of the prelle singer ps method a method used to find first integrals to such systems in order to reduce their order we consider the cases of coupled second order odes and coupled third order odes as well a

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

integrability properties of a generalized reduction of the kdv6 equation abstract we consider the integrability properties of a generalized version of a similarity reduction of the so called kdv6 equation an equation that has recently generated much interest we give a linear problem for this generalized reduction and show that it satisfies the requirements of the ablowitz ramani segur algorithm in addition we give a backlund transformation to a related equation giving also an auto backlund transformation for this last our results mirror those for the korteweg de vries equation itself which has a similarity reduction to an ordinary differential equation which is related by a backlund transformation to the second painleve equation this last having an auto backlund transformation

auto backlund transformations and integrability of ordinary and partial differential equations we introduce a partial differential equation whose structure in many respects mimics that of the second painleve equation for this equation we give auto backlund transformations of both ordinary differential equation and partial differential equation types our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto backlund transformations and the integrability of ordinary and partial differential equations

on a construction of general formal solutions for equations of the first painleve hierarchy i abstract we construct general formal solutions containing sufficiently many free parameters for the first painleve hierarchy with a large parameter from a viewpoint of the multiple scale analysis

systematic method of generating new integrable systems via inverse miura maps we provide a new natural interpretation of the lax representation for an integrable system that is the spectral problem is the linearized form of a miura transformation between the original system and a modified version of it on the basis of this interpretation we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by miura transformations thus this method can be used to generate new integrable systems from known systems through inverse miura maps it can be applied to both continuous and discrete systems in 1 1 dimensions as well as in 2 1 dimensions the effectiveness of the method is illustrated using examples such as the nonlinear schrodinger nls system the zakharov ito system two component kdv the three wave interaction system the yajima oikawa system the ablowitz ladik lattice integrable space discrete nls and two 2 1 dimensional nls systems

on an extended second painleve hierarchy abstract we present a new extension of the second painleve hierarchy and study its properties in addition to lax pairs backlund transformations auto backlund transformations and basic special integrals we also consider a new phenomenon whereby we obtain relations between systems of different orders but of the same form the extension made here of the second painleve hierarchy is based on the use of non isospectral scattering problems and so is quite general we thus expect to be able to obtain similar extensions of other painleve hierarchies including not only for continuous examples but also for discrete and differential delay examples we believe that our work is also of relevance for painleve classification since it gives information about classes of equation that may be of interest and in addition provides a key to the possible identification of equations isolated in such a process

matrix completion by least square low rank and sparse self representations abstract conventional matrix completion methods are generally based on rank minimization these methods assume that the given matrix is of low rank and the data points are drawn from a single subspace of low dimensionality therefore they are not effective in completing matrices where the data are drawn from multiple subspaces in this paper we establish a novel matrix completion framework that is based on self representation specifically least square low rank and sparse self representations based matrix completion algorithms are provided the underlying idea is that one data point can be efficiently reconstructed by other data points belonging to a common subspace where the missing entries are recovered so as to fit the common subspace the proposed algorithms actually maximize the weighted correlations among the columns of a given matrix we prove that the proposed algorithms are approximations for rank minimization of the incomplete matrix in addition they are able to complete high rank or even full rank matrices when the data are drawn from multiple subspaces comparative studies are conducted on synthetic datasets natural image inpainting tasks temperature prediction task and collaborative filtering tasks the results show that the proposed algorithms often outperform other state of the art algorithms in various tasks

symmetries and hierarchies of equations for the 2 1 dimensional n wave interaction abstract infinite sets of symmetries for the 2 1 dimensional n wave interaction are derived and two hierarchies of equations are given the lax pairs for these hierarchies are derived by an algebraic approach consequently two hierarchies are proved to be respectively the isospectral and non isospectral flows thus the inverse scattering transform of the multidimensional akns spectral problem can be applied and the corresponding time evolution of scattering data are obtained

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

on closed form solutions of 2 1 breaking soliton system by similarity transformations method abstract in the present research similarity transformation method via lie group theory is proposed to seek some more exact closed form solutions of the 2 1 dimensional breaking soliton system the system describes the interactions of the riemann wave along y axis and long wave along x axis some explicit solutions of breaking soliton system are attained with appropriate choices of the arbitrary functions and making use of arbitrary constants involved in the infinitesimals in order to obtain physically meaningful solutions numerical simulation is performed on the basis of graphical representation the physical analysis of solutions reveals into multi solitons periodic quadratic asymptotic and stationary profiles

2 1 dimensional integrable lattice hierarchies related to discrete fourth order nonisospectral problems in this paper we consider the 2 1 dimensional discrete fourth order nonisospectral problem by using the lax technique three new 2 1 dimensional nonisospectral four field integrable lattice hierarchies are constructed their reductions yield three 1 1 dimensional isospectral four field integrable lattice hierarchies due to mlaszak marciniak we make a comparison between the 1 1 dimensional discrete fourth order nonisospectral problem and the third order nonisospectral problem we found that the integrable lattice hierarchies related to the discrete fourth order nonisospectral problem have new characteristics

a method of reduction of order for discrete systems we show how by establishing a suitable backlund correspondence a reduction in order can be effected for discrete systems having a certain structure in this way we succeed in extending to discrete systems ideas previously developed within the context of ordinary differential equations it is also worth emphasizing that our approach represents an application of properties characteristic of completely integrable systems outside of that context that is to systems of discrete equations integrable or otherwise

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

algebraic geometrical solutions of some multidimensional nonlinear evolution equations the known 2 1 dimensional breaking soliton equation the coupled kp equation with three potentials and a new 3 1 dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the 1 1 dimensional akns equations the abel jacobi coordinates are introduced to straighten out the associated flows from which algebraic geometrical solutions of the 2 1 dimensional breaking soliton equation the coupled kp equation and the 3 1 dimensional evolution equation are explicitly given in terms of the riemann theta functions

amalgamations of the painleve equations new hierarchies of nonlinear ordinary differential equations that are generalizations of the painleve equations are presented these hierarchies are shown to contain the painleve equations as special cases emphasis is on the sixth order ordinary differential equations special solutions for one of them are expressed via the general solutions of the p1 p2 equations and special cases of the p3 and p5 equations special solutions of another sixth order ode are determined by the general solutions of the p2 p4 p3 and p5 equations five of six painleve equations can be considered as special cases of these sixth order odes isomonodromic linear problems to solve the cauchy problems of hierarchies by the inverse monodromy transform are given

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

symmetries exact solutions and nonlinear superposition formulas for two integrable partial differential equations we recently introduced two new sixth order partial differential equations pdes associated with third order scattering problems here we extend our study of these pdes by considering the construction of exact solutions both by using the method of symmetry reduction due to lie and by using their darboux transformations dts amongst the ordinary differential equations odes obtained by symmetry reduction is an ode due to cosgrove that is believed to define a new painleve transcendent this ode provides soliton solutions for our integrable pdes that include arbitrary functions of time the dts for our pdes allow the recovery of these solutions and in addition provide other solutions which are not associated with lie symmetries either classical or nonclassical we also consider the iteration of the corresponding backlund transformations bts for these pdes the theorem of permutability allows us to reduce this process of iterating the dt from one of solving a third order linear equation the spatial p

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

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

commutator subgroup of vershik kerov group ii abstract we describe a commutator subgroup of vershik kerov group over an infinite field and find the bound for its commutator width this gives a partial solution of the problem posed by sushchanskii in 2010 we also describe the lower central series of the group of infinite upper triangular matrices over an infinite field and find the bound for its commutator width

commutators and powers of infinite unitriangular matrices abstract in the paper we consider some commutator type and power type matrix equations in the group ut k of infinite dimensional unitriangular matrices over a field k we introduce a notion of a power outer commutator k m 1 m k x 1 x k and a power engel commutator e k l m 1 m k x y as outer respectively engel commutators modified by allowing powers of letters instead of letters alone for a given infinite unitriangular matrix a we discuss the matrix equations x k a k m 1 m k x 1 x k a and e k l m 1 m k x y a in variables x x 1 x k y as a main result we provide the necessary and sufficient conditions for solvability of these equations

low rank matrix fitting based on subspace perturbation analysis with applications to structure from motion the task of finding a low rank r matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering the problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements the former problem can be solved by concatenating a set of r column matrices that share a common single r dimensional solution space unfortunately the number of possible submatrices is generally very large and hence the results obtained with one set of r column matrices will generally be different from that captured by a different set ideally we would like to find that solution that is least affected by noise this requires that we determine which of the r column matrices i e which of the original feature points are less influenced by the unknown noise term this paper presents a criterion to successfully carry out such a selection our key result is to formally prove that the more distinct the r vectors of the r column matrices are the less they are swayed by noise this key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r column matrices it is shown how this criterion can be effectively used to recover the noise free matrix of rank r finally we derive the affine and projective structure from motion sfm algorithms using the proposed criterion extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

using symbolic computation to exactly solve for the bogoyavlenskii s generalized breaking soliton equation abstract in this paper the symbolic computation systems help us find some exact solutions for the bogoyavlenskii s generalized breaking soliton equation which are the sums of the arbitrary functions of time and parameterized solitary waves

schlesinger transformations for the second members of pii and piv hierarchies in this paper we give a method to obtain the schlesinger transformations for the second members of second and fourth painlev e hierarchies the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

higher dimensional painleve integrable models from the kadomtsev petviashvili equation after embedding the kadomtsev petviashvili equation in higher dimensions and extending the painleve analysis approach to a new form such that the coefficients of the expansion around the singular manifold possess conformal invariance and contain explicit new space variables we can get infinitely many painleve integrable models in 3 1 dimensions and higher dimensions some concrete higher dimensional modified korteweg de vries type of extensions are given whether the models are lax integrable or integrable under other meanings remain still open

parabolic subgroups of groups of column finite infinite matrices abstract for a wide class of rings r that contains all local and semilocal rings we consider the group sl vk r of all matrices over r of the form a b 0 c in which a is a finite matrix with determinant 1 and c is an upper triangular infinite matrix with the main diagonal 1 1 we describe all subgroups of sl vk r that contain all its upper triangular matrices and study their properties

new 2 1 dimensional nonisospectral toda lattice hierarchy we give a new 2 1 dimensional nonisospectral generalization of the toda lattice hierarchy reductions yield a variety of new integrable hierarchies along with their underlying linear problems including new 1 1 dimensional differential delay hierarchies nonisospectral and isospectral new ordinary differential delay hierarchies and new discrete painleve hierarchies we also show that a reduction in components yields our previously obtained 2 1 dimensional nonisospectral volterra lattice hierarchy

pole free solutions of the first painleve hierarchy and non generic critical behavior for the kdv equation we establish the existence of real pole free solutions to all even members of the painleve i hierarchy we also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the kdv equation in the small dispersion limit this was understood in the case of a generic critical point and we generalize it here to the case of non generic critical points

matrix semidiscrete ablowitz ladik equation hierarchy and a matrix discrete second painleve equation in this article we first show that the semidiscrete ablowitz ladik equation hierarchy can be extended to the matrix case by considering a matrix semidiscrete ablowitz ladik isospectral linear problem we then extend this linear problem to the nonisospectral case and a new coupled matrix semidiscrete integrable equation is constructed it is shown that the continuum limit of this new matrix semidiscrete integrable equation is the coupled matrix variable coefficient modified korteweg de vries equation and that a matrix discrete second painleve equation can be obtained as a stationary reduction in this new matrix semidiscrete equation

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

a special solution technique further extensions abstract in a recent paper we gave a technique for identifying special solutions of hierarchies of ordinary differential equations odes and difference equations in particular we considered the phenomenon whereby members of one painleve hierarchy define solutions of members of a different painleve hierarchy where both are related to the same at least bi hamiltonian completely integrable partial differential equation pde or lattice hierarchy we also considered the question of special integrals of further related painleve hierarchies here we consider three extensions of this approach first of all we consider the case of ode hierarchies where the corresponding pde hierarchy is only mono hamiltonian second we show that our approach can also be extended in order to provide special solutions of pdes our third extension is to the non hamiltonian case where we consider hierarchies of odes related to the burgers hierarchy we also make the observation that our results can be phrased quite generally and the systems of equations dealt with do not in fact need to be integrable

wkb analysis of higher order painleve equations with a large parameter local reduction of 0 parameter solutions for painleve hierarchies pj j i ii 1 or ii 2 abstract we generalize the reduction theorem for 0 parameter solutions of a traditional i e second order painleve equation with a large parameter to those of some higher order painleve equation that is each member of the painleve hierarchies p j j i ii 1 or ii 2 thus the scope of applicability of the reduction theorem in kt1 has been substantially enlarged only six equations were covered by our previous result while theorem 3 2 1 of this paper applies to infinitely many equations

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

a fast tri factorization method for low rank matrix recovery and completion in recent years matrix rank minimization problems have received a significant amount of attention in machine learning data mining and computer vision communities and these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix and has to be solved iteratively and involves singular value decomposition svd at each iteration therefore those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple svds in this paper we propose a fast tri factorization ftf method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing svds the proposed ftf method can be used to reliably solve a wide range of low rank matrix recovery and completion problems such as robust principal component analysis rpca low rank representation lrr and low rank matrix completion mc we also present three specific models for rpca lrr and mc problems respectively moreover we develop two alternating direction method adm based iterative algorithms for solving the above three problems experimental results on a variety of synthetic and real world data sets validate the efficiency robustness and effectiveness of our ftf method comparing with the state of the art nuclear norm minimization algorithms

lattice modified kdv hierarchy from a lax pair expansion we produce a hierarchy of integrable equations by systematically adding terms to the lax pair for the lattice modified kdv equation the equations in the hierarchy are related to one another by recursion relations these recursion relations are solved explicitly so that every equation in the hierarchy along with its lax pair is known

products of three triangular matrices abstract we prove that any square matrix over a field is the product of at most three triangular matrices we also give explicit lul factorizations for all 2 2 and 3 3 matrices over fields

the motion of surfaces in geodesic coordinates and 2 1 dimensional breaking soliton equation the system of evolution equations for general motion of surfaces in geodesic coordinates is analyzed to reduce the number of variables as well as equations a special choice for some variants deduces that the surface and the motion of surface correspond to a nonlinear schrodinger equation and a 2 1 dimensional breaking soliton equation respectively we also study some geometric properties corresponding to the integrals of the motion of surface and the motion of the curve on the surface the spectral parameter is introduced

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

on a new hierarchy of symmetries for the kadomtsev petviashvili equation abstract we present a new hierarchy of symmetries for the kadomtsev petviashvili equation these new symmetries depend on the space and time variables explicitly together with the previously known classical symmetries they constitute an infinite dimensional lie algebra

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy