a super discrete variational identity and its application for constructing super discrete hamiltonian systems in the paper a super discrete variational identity on lie super algebras is established first it provides an approach for constructing super discrete hamiltonian structures of super evolution lattice equations with discrete zero curvature representation when the super spectral matrix u is selected appropriately as an application super discrete hamiltonian structures of super toda lattice hierarchy are developed and this method can be used to construct super discrete bi hamiltonian structures of more super discrete integrable evolutive hierarchies super discrete variational identity is devoted to deduce super integrability and solutions of super discrete integrable equations

the liouville integrable systems associated with a new discrete 3 3 matrix spectral problem a new discrete 3 times 3 matrix spectral problem with three potentials is introduced and the corresponding family of liouville integrable lattice equations is obtained by applying the discrete trace identity it is shown that the hierarchy possesses a hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals finally infinitely many conservation laws of corresponding lattice systems are obtained by a direct way

continuous limits for an integrable coupling of the kac van moerbeke hierarchy in this paper we present an integrable coupling of lattice hierarchy and its continuous limits by using lie algebra sl 4 by introducing a complex discrete spectral problem the integrable coupling of kac van moerbeke lattice hierarchy is derived it is shown that a new complex lattice spectral problem converges to the integrable coupling of discrete soliton equation hierarchy which has the integrable coupling system of mkdv hierarchy as a new kind of continuous limit

integrable coupling of the ablowitz ladik hierarchy and its hamiltonian structure abstract the trace identity is generalized to work for the discrete zero curvature equation associated with the lie algebra possessing degenerate killing forms then a kind of integrable coupling of the ablowitz ladik al hierarchy is obtained and its hamiltonian structure is worked out moreover liouville integrability of the integrable coupling is demonstrated

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

a bi hamiltonian formulation for triangular systems by perturbations a bi hamiltonian formulation is proposed for triangular systems resulting from perturbations around solutions from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed provided that one operator of the hamiltonian pair is invertible through our formulation four examples of triangular systems are exhibited which also show that bi hamiltonian systems in both lower dimensions and higher dimensions are many and varied two of four examples give local 2 1 dimensional bi hamiltonian systems and illustrate that multiscale perturbations can lead to higher dimensional bi hamiltonian systems

integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of lie algebras abstract staring from a new spectral problem a hierarchy of the soliton equations is derived it is shown that the associated hierarchies are infinite dimensional integrable hamiltonian systems by the procedure of nonlinearization of the lax pairs the integrable decomposition of the whole soliton hierarchy is given further we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of lie algebras

hamiltonian and super hamiltonian systems of a hierarchy of soliton equations abstract by considering an isospectral eigenvalue problem a hierarchy of soliton equations are derived two types of extensions are presented by enlarging the associated spectral problem with the aid of generalized trace identity and the super trace identity the hamiltonian and super hamiltonian structures for the integrable extensions are established

an integrable system and associated integrable models as well as hamiltonian structures starting from an existed lie algebra introduces a new lie algebra a1 e1 e2 e3 so that two isospectral lax matrices are established by employing the tu scheme an integrable equation hierarchy denoted by ieh is obtained from which a few reduced evolution equations are presented one of them is the mkdv equation the elliptic variable solutions and three kinds of darboux transformations for one coupled equation which is from the ieh are worked out respectively finally we take use of the lie algebra a1 to generate eight higher dimensional lie algebras from which the linear integrable couplings the nonlinear integrable couplings and the bi integrable couplings of the ieh are engendered whose hamiltonian structures are also obtained by the variational identity then further reduce one coupled integrable equation to get a nonlinear generalized mkdv equation

the integrable coupling system of a 3 3 discrete matrix spectral problem integrable coupling with six potentials is first proposed by coupling a given 3x3 discrete matrix spectral problem it is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators which yield infinitely many commuting symmetries moreover by means of the discrete variational identity on semi direct sums of lie algebras the hamiltonian form is deduced for the lattice equations in the resulting hierarchy finally we prove that the hierarchy of the resulting hamiltonian equations is liouville integrable discrete hamiltonian system

new super integrable hierarchies associated with osp 2 2 and spo 2 2 and their applications abstract based on the lie superalgebras osp 2 2 and spo 2 2 we generate some new super soliton hierarchies these hierarchies can reduce to the akns hierarchy and the super akns hierarchy and by using the supertrace identity on lie superalgebras we construct the super hamiltonian structures of these new super soliton hierarchies then as the applications of spo 2 2 osp 2 2 and spo 2 2 sl 1 2 we obtain the relations of different hierarchies associated with different lie superalgebras

new non isospectral integrable couplings of the akns system abstract by means of semi direct sums of lie algebras and the compatibility condition of lax pair an efficient way to construct non isospectral integrable couplings is proposed as its applications new non isospectral integrable couplings of the akns system are obtained the method can be devoted to constructing non isospectral integrable couplings of other nonlinear evolution equations

an integrable couplings of g wki equations hierarchy with self consistent sources we present a kind of integrable couplings of soliton equations hierarchy with self consistent sources by using the loop algebra sl 4 as an application a new integrable couplings of the generalized wadati konono ichikawa g wki equations hierarchy with self consistent sources is derived through the enlarged loop algebra sl 4

nonlinear continuous integrable hamiltonian couplings based on a kind of special non semisimple lie algebras a scheme is presented for constructing nonlinear continuous integrable couplings variational identities over the corresponding loop algebras are used to furnish hamiltonian structures for the resulting continuous integrable couplings the application of the scheme is illustrated by an example of nonlinear continuous integrable hamiltonian couplings of the akns hierarchy of soliton equations

nonlinear super integrable hamiltonian couplings based on a kind of new lie superalgebras we proposed a scheme for constructing nonlinear super integrable couplings super trace identity over the corresponding loop superalgebras is used to furnish the super hamiltonian structures for the resulting nonlinear super integrable couplings we illustrate the application of the scheme by means of an enlarged super ablowitz kaup newell segur akns matrix spectral problem and present a nonlinear super integrable hamiltonian couplings for the super akns hierarchy

the generalized fractional trace variational identity and fractional integrable couplings of kaup newell hierarchy starting from fractional isospectral problems and general bilinear forms the generalized fractional trace identity is presented then a new explicit lie algebra is introduced for which the fractional integrable couplings of kaup newell hierarchy are derived furthermore we obtained the fractional hamiltonian structures of the fractional integrable couplings of kaup newell hierarchy the methods derived by us can be generalized to other fractional integrable couplings

a new integrable symplectic map associated with lattice soliton equations a method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3 3 matrix spectral problem a new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of lax pairs and their adjoint lax pairs of the hierarchy moreover the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map

pyramid segmentation algorithms revisited the main goal of this work is to compare pyramidal structures proposed to solve segmentation tasks segmentation algorithms based on regular and irregular pyramids are described together with the data structures and decimation procedures which encode and manage the information in the pyramid in order to compare the different segmentation algorithms we have employed three types of quality measurements the shift variance measure the f function and the q function

a spectral problem based on so 3 r and its associated commuting soliton equations we present a spectral problem based on the three dimensional real special orthogonal lie algebra so 3 r and construct a hierarchy of commuting bi hamiltonian soliton equations by zero curvature equations associated with the spectral problem an illustrative example of soliton equations is computed together with its associated bi hamiltonian structure

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

new form 2 1 dimensional integrable lattice hierarchies with two shift operators e and eˆ abstract we present a new method to construct two 2 1 dimensional lattice hierarchies by considering a kind of spectral problems with two shift operators e and e a new 2 1 dimensional discrete soliton hierarchy and a new 3 3 lattice hierarchy are constructed by using 3 3 matrix spectral problem

a new loop algebra and a corresponding integrable hierarchy as well as its integrable coupling a type of new interesting loop algebra gm m 1 2 with a simple commutation operation just like that in the loop algebra a1 is constructed with the help of the loop algebra gm a new multicomponent integrable system m akns kn hierarchy is worked out as reduction cases the m akns hierarchy and m kn hierarchy are engendered respectively in addition the system 1 akns kn which is a reduced case of the m akns kn hierarchy above is a unified expressing integrable model of the akns hierarchy and the kn hierarchy obviously the m akns kn hierarchy is again a united expressing integrable model of the multicomponent akns hierarchy m akns and the multicomponent kn hierarchy m kn this article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations finally we work out an integrable coupling of the m akns kn hierarchy

an integrable coupling family of the toda lattice systems its bi hamiltonian structure and a related nonisospectral integrable lattice family starting from a four by four matrix spectral problem an integrable coupling family of the toda lattice systems is derived a pair of discrete hamiltonian operators is presented and a sequence of the corresponding hamiltonian functionals is constructed by using the discrete variational identity then the bi hamiltonian structure of the obtained integrable coupling family is established finally a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

constructing nonlinear discrete integrable hamiltonian couplings beginning with lax pairs from special non semisimple matrix lie algebras we establish a scheme for constructing nonlinear discrete integrable couplings discrete variational identities over the associated loop algebras are used to build hamiltonian structures for the resulting integrable couplings we illustrate the application of the scheme by means of an enlarged volterra spectral problem and present an example of nonlinear discrete integrable hamiltonian couplings for the volterra lattice equations

numerical solution of fully nonlinear elliptic equations by bohmer s method abstract we present an implementation of bohmer s finite element method for fully nonlinear elliptic partial differential equations on convex polygonal domains based on a modified argyris element and bernstein bezier techniques our numerical experiments for several test problems involving the classical monge ampere equation and an unconditionally elliptic equation confirm the convergence and error bounds predicted by bohmer s theoretical results

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

the quadratic form identity for constructing the hamiltonian structures of the discrete integrable systems the quadratic form identity is extended to the discrete version which can be used to construct the hamiltonian structures of the discrete integrable systems associated with the lie algebra possessing degenerate killing forms especially it can be used to work out the hamiltonian structures of some kinds of discrete integrable couplings then a kind of integrable coupling of the toda hierarchy is obtained and its hamiltonian structure is worked out by using the discrete quadratic form identity moreover the liouville integrability of the integrable coupling is demonstrated

integrable semi discretization of the coupled modified kdv equations the discrete version of the inverse scattering method proposed by ablowitz and ladik is extended to solve multi component systems the extension enables one to solve the initial value problem which proves directly the complete integrability of a semi discrete version of the coupled modified korteweg de vries kdv equations and their hierarchy it also provides a procedure to obtain conservation laws and multi soliton solutions of the hierarchy

two integrable couplings of the tu hierarchy and their hamiltonian structures the double integrable couplings of the tu hierarchy are worked out by use of vector loop algebras g 6 and g 9 respectively also the hamiltonian structures of the obtained system are given by the quadratic form identity

constructing integrable coupling systems of hamiltonian lattice equations using semi direct sums of lie algebras abstract in this article by considering a discrete isospectral problem a hierarchy of hamiltonian lattice equations are derived two types of semi direct sums of lie algebras are proposed using which a practicable way to construct discrete integrable couplings is introduced as an application two kinds of discrete integrable couplings of the resulting system are worked out

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

two integrable couplings of a generalized d kaup newell hierarchy and their hamiltonian and bi hamiltonian structures abstract we enlarge the spectral problem of a generalized d kaup newell d kn spectral problem solving the enlarged zero curvature equations we produce integrable couplings a reduction of the spectral matrix leads to a second integrable coupling system next bilinear forms that are symmetric ad invariant and non degenerate on the given non semisimple matrix lie algebra are computed to employ the variational identity the variational identity is then applied to the original enlarged spectral problem of a generalized d kn hierarchy and the reduced problem hamiltonian structures are presented as well as a bi hamiltonian formulation of the reduced problem both hierarchies have infinitely many commuting high order symmetries and conserved densities i e are liouville integrable

adaptive combination of pca and vq networks in this paper we consider the principal component analysis pca and vector quantization vq neural networks for image compression we present a method where the pca and vq steps are adaptively combined a learning algorithm for this combined network is derived we demonstrate that this approach can improve the results of the successive application of the individually optimal methods

an integrable lattice hierarchy associated integrable coupling darboux transformation and conservation laws abstract a new integrable lattice hierarchy is constructed from a discrete matrix spectral problem some related properties of the new hierarchy are discussed the hamiltonian structures and liouville integrability of the new hierarchy are established by using the discrete trace identity a kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems a darboux transformation dt with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its lax representation the explicit solutions are obtained via the dt the structures for those solutions are graphically investigated all these properties might be helpful to understanding some physical phenomena

a family of liouville integrable lattice equations and its conservation laws abstract a new discrete spectral problem of matrix three by three with three potentials is introduced and by using the discrete trace identity a hierarchy of liouville lattice equations is obtained the infinite number of conservation laws of the hierarchy of hamiltonian system are presented

a generalized ablowitz ladik hierarchy multi hamiltonian structure and darboux transformation starting from a discrete spectral problem we derive a hierarchy of nonlinear discrete equations which includes the ablowitz ladik hierarchy and a new hierarchy as special cases especially we investigate in detail the integrability and resolvability of the new hierarchy it is shown that the new hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure a darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its lax pair as applications of the darboux transformation new exact solutions of the discrete system are explicitly given

four lie algebras associated with r6 and their applications the first part in the paper reads that a three dimensional lie algebra is first introduced whose corresponding loop algebra is constructed for which isospectral problems are established by employing zero curvature equations a modified kaup newell mkn soliton hierarchy of evolution equations is obtained the corresponding hereditary operator and hamiltonian structure are worked out respectively then two types of enlarging semisimple lie algebras isomorphic to the linear space r6 are followed to construct one of them is a complex lie algebra their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mkn hierarchy the hereditary operators hamiltonian structures of the hierarchies are produced again respectively the exact computing formulas of the constant appearing in the trace identity and the variational identity are derived under the semisimple algebras the second part of this pap

integrable couplings of vector akns soliton equations by enlarging the associated matrix spectral problems two specific classes of multicomponent integrable couplings of the physically important vector akns soliton equations are constructed which can be linked to each other by a backlund transformation the resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures and thus each system in the two hierarchies has infinitely many commuting symmetries

a soliton hierarchy associated with so 3 r we generate a hierarchy of soliton equations from zero curvature equations associated with the real lie algebra so 3 r and show that each equation in the resulting hierarchy has a bi hamiltonian structure and thus integrable in the liouville sense

weighted least squares training of support vector classifiers leading to compact and adaptive schemes an iterative block training method for support vector classifiers svcs based on weighted least squares wls optimization is presented the algorithm which minimizes structural risk in the primal space is applicable to both linear and nonlinear machines in some nonlinear cases it is necessary to previously find a projection of data onto an intermediate dimensional space by means of either principal component analysis or clustering techniques the proposed approach yields very compact machines the complexity reduction with respect to the svc solution is especially notable in problems with highly overlapped classes furthermore the formulation in terms of wls minimization makes the development of adaptive svcs straightforward opening up new fields of application for this type of model mainly online processing of large amounts of static stationary data as well as online update in nonstationary scenarios adaptive solutions the performance of this new type of algorithm is analyzed by means of several simulations

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

the generalized kupershmidt deformation for the fifth order coupled kdv equations hierarchy abstract based on the kupershmidt deformation we propose the generalized kupershmidt deformation gkd to construct new systems from integrable bi hamiltonian system as applications the generalized kupershmidt deformation of the fifth order coupled kdv equations hierarchy with self consistent sources and its lax representation are presented

factorization of a hierarchy of the lattice soliton equations from a binary bargmann symmetry constraint abstract staring from a discrete spectral problem a hierarchy of the lattice soliton equations is derived it is shown that each lattice equation in resulting hierarchy is liouville integrable discrete hamiltonian system the binary nonlinearization of the lax pairs and the adjoint lax pairs of the resulting hierarchy is discussed each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite dimensional integrable system in liouville sense especially factorization of a discrete kdv equation is given

new lax pairs of the toda lattice and the nonlinearization under a higher order bargmann constraint by starting from a new discrete spectral problem the toda lattice is derived through the discrete zero curvature equation applying the discrete variational identity to the spectral problem will also reach to the bi hamiltonian structure of the toda lattice under a higher order bargmann symmetry constraint the new lax pairs and the adjoint lax pairs are nonlinearized into integrable symplectic maps and finite dimensional liouville integrable hamiltonian systems finally a backlund transformation of the toda lattice is obtained

hamiltonian structures and conservation laws of the belov chaltikian lattice hierarchy a hierarchy of nonlinear differential difference equations associated with a 3 3 matrix spectral problem is proposed through discrete zero curvature equations one of the nontrivial equations in this hierarchy is the famous belov chaltikian lattice equation it is shown that the hierarchy possesses infinitely many conservation laws hamiltonian structures for this hierarchy are constructed with the aid of the trace identity

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

symmetry constraint of mkdv equations by binary nonlinearization abstract a symmetry constraint for mkdv integrable hierarchy is presented by binary nonlinearization the spatial part and the temporal parts of the lax pairs and the adjoint lax pairs of mkdv equations are all constrained as finite dimensional liouville integrable hamiltonian systems whose integrals of motion are explicitly given in terms of the proposed symmetry constraint mkdv equations are decomposed into two finite dimensional liouville integrable constrained systems and thus a kind of separation of variables for mkdv equations is established

an integrable couplings of dirac soliton hierarchy with self consistent sources abstract a kind of the integrable couplings of soliton equations hierarchy with self consistent sources associated with sl 4 is presented as an application example we construct a new integrable couplings of the dirac soliton hierarchy with self consistent sources by using of loop algebra sl 4

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

a non isospectral integrable couplings of volterra lattice hierarchy with self consistent sources a kind of non isospectral integrable couplings of discrete soliton equations hierarchy with self consistent sources associated with email protected 4 y f zhang e g fan characteristic numbers of matrix lie algebras commun theor phys china 49 2008 845 is presented as an application example the hierarchy of non isospectral cubic volterra lattice hierarchy with self consistent sources is derived furthermore we construct a non isospectral integrable couplings of cubic volterra lattice hierarchy with self consistent sources by using the loop algebra email protected 4

a deformed reduced semi discrete kaup newell equation the related integrable family and darboux transformation we propose a deformed reduced semi discrete kaup newell equation we derive the related integrable family through discrete zero curvature equation we establish a darboux transformation of lax pair of the proposed equation three exact solutions are given a deformed reduced semi discrete kaup newell equation and its related integrable family are derived from discrete zero curvature equation a darboux transformation of lax pair of this equation is established with the help of gauge transformation by means of the resulting darboux transformation three exact solutions are given

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

nonlinearization of spectral problems for the perturbation kdv systems abstract the theory of nonlinearization of spectral problems is developed for investigating the perturbation kdv systems and a kind of finite dimensional integrable hamiltonian systems is generated from spectral problems of the perturbation kdv systems to establish the liouville integrability of the obtained hamiltonian systems we identify them with the perturbation systems of the garnier system which can be directly proved to be integrable a useful formula to compute the functional gradient of the spectral parameter with respect to the potential is also presented for arbitrary order matrix spectral problems

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

on finite element methods for fully nonlinear elliptic equations of second order for the first time we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard c 1 finite element method fem we consider throughout the paper two cases in parallel for convex bounded polyhedral domains in bf r n or for c 2 bounded domains in bf r 2 we prove stability and convergence for the corresponding conforming or nonconforming c 1 fem respectively the results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere the classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator the consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated the stability has to be proved in an unusual way this is the hard core of the paper essential tools are linearization a compactness argument the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations an essential basis for our proofs are davydov s results for c 1 fes on polyhedral domains in bf r n or of local degree 5 for c 2 domains in bf r 2 better convergence and extensions to bf r n for c 2 domains are to be expected from his forthcoming results on curved domains our proof for the second case in bf r n includes the first essentially as a special case the method applies to quasi linear elliptic problems not in divergence form as well a discrete newton method is shown to converge locally quadratically essentially independently of the actual grid size by the mesh independence principle

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

a nonlinear integrable couplings of c kdv soliton hierarchy and its infinite conservation laws abstract a kind of nonlinear integrable couplings of c kdv soliton hierarchy is constructed then we present the infinitely many conservation laws for the nonlinear integrable couplings of the c kdv soliton hierarchy

a finite element method for nonlinear elliptic problems we present a galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations a key tool is the discretization proposed in lakkis and pryer 2011 allowing us to work directly on the strong form of a linear pde an added benefit to making use of this discretization method is that a recovered finite element hessian is a byproduct of the solution process we build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear pdes benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the monge ampere equation and the pucci equation

a lie algebra containing four parameters for the generalized dirac hierarchy a lie algebra containing four parameters is obtained whose commutation operation is concise and the corresponding computing formula of constant c in the variational identity is presented in this paper as application a new liouville integrable hierarchy which can be reduced to dirac hierarchy is derived by designing a special isospectral problem we call it generalized dirac hierarchy

c 0 penalty methods for the fully nonlinear monge ampere equation in this paper we develop and analyze c 0 penalty methods for the fully nonlinear monge ampere equation det d 2 u f in two dimensions the key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric stable and consistent with the continuous linearization we are then able to show the well posedness of the penalty method as well as quasi optimal error estimates using the banach fixed point theorem as our main tool numerical experiments are presented which support the theoretical results

hierarchies integrable decompositions and conservation laws of geng equation abstract in this paper two hierarchies of the geng equations are presented including positive non isospectral hierarchy and negative non isospectral hierarchy moreover integrable couplings of the corresponding geng hierarchies are also constructed by enlarging the associated matrix spectral problem three new integrable decompositions and conservation laws of the isospectral geng equation are also obtained the gauge transformations are used to obtain the associated binary symmetry constraints of the geng equation at the first time

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

integrable couplings of relativistic toda lattice systems in polynomial form and rational form their hierarchies and bi hamiltonian structures integrable couplings of relativistic toda lattice systems in polynomial form and rational form and their hierarchies are derived from a four by four discrete matrix eigenvalue problem the bi hamiltonian structure for every integrable coupling in the two hierarchies obtained is established by means of the discrete variational identity ultimately liouvolle integrability of the obtained integrable couplings is demonstrated

binary symmetry constraints of n wave interaction equations in 1 1 and 2 1 dimensions binary symmetry constraints of the n wave interaction equations in 1 1 and 2 1 dimensions are proposed to reduce the n wave interaction equations into finite dimensional liouville integrable systems a new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix lax operator and used to show the liouville integrability of the constrained flows of the n wave interaction equations the constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the n wave interaction equations and thus the integrability by quadratures are shown for the n wave interaction equations by the constrained flows

completion of the ablowitz kaup newell segur integrable coupling integrable couplings are associated with non semisimple lie algebras in this paper we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings which is called the completion process of integrable couplings as an example the idea of construction is applied to the ablowitz kaup newell segur integrable coupling each equation in the resulting hierarchy has a bi hamiltonian structure furnished by the component trace identity integrable couplings are associated with non semisimple lie algebras in this paper we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings which is called the completion process of integrable couplings as an example the idea of construction is applied to the ablowitz kaup newell segur integrable coupling each equation in the resulting hierarchy has a bi hamiltonian structure furnished by the component trace identity

finite element methods for fully nonlinear second order pdes based on a discrete hessian with applications to the monge ampere equation the purpose of this paper is twofold first we modify a method due to lakkis and pryer where the notion of a discrete hessian is introduced to compute fully nonlinear second order pdes the discrete hessian used in our approach is entirely local making the resulting linear system within the newton iteration much easier to solve the second contribution of this paper is to analyze both lakkis and pryer s method and its modification in parallel applied to the two dimensional monge ampere equation in both cases we show the well posedness of the methods as well as derive optimal error estimates numerical experiments are presented which i back up the theoretical findings and ii indicate that the methods are able to capture weak viscosity solutions

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed