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

symmetries and constants of motion for new hierarchy of the kp equations abstract two sets of infinite numbers of symmetries and constants of motion for a class of 2 1 dimensional nonlinear evolution equations nees are derived these nees we call them the new hierarchy of the kp equations are generated by a set of additional symmetries of the kp and higher order kp equations and explicitly depend on the space variables x and y the expressions for their constants of motion in terms of the scattering data are given and the time evolution of the scattering data corresponding to the new hierarchy of the kp equations are computed

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

binary nonlinearization of the super akns system under an implicit symmetry constraint for the super akns system an implicit symmetry constraint between the potentials and the eigenfunctions is proposed after introducing some new variables to explicitly express potentials the super akns system is decomposed into two compatible finite dimensional super systems x part and tn part furthermore we show that the obtained super systems are integrable super hamiltonian systems in the supersymmetry manifold

the lie algebraic structure of symmetries generated by hereditary symmetries based on the works of li zhu and co workers 1985 a general lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries

the bi hamiltonian structure of fully supersymmetric korteweg de vries systems the bi hamiltonian structure of integrable supersymmetric extensions of the korteweg de vries kdv equation related to then 1 and then 2 superconformal algebras is found it turns out that some of these extensions admit inverse hamiltonian formulations in terms of presymplectic operators rather than in terms of poisson tensors for one extension related to then 2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical kdv they can be explained by a factorization of the corresponding lax operator all the bi hamiltonian formulations are derived in a systematic way from the lax operators

supersymmetric kadomtsev petviashvili hierarchy ghost symmetry structure reductions and darboux backlund solutions this paper studies manin radul supersymmetric kadomtsev petviashvili hierarchy mr skp in three related aspects i we find an infinite set of additional ghost symmetry flows spanning the same anti commutation algebra as the ordinary mr skp flows ii the latter are used to construct consistent reductions skpr 2 m 2 of the initial unconstrained mr skp hierarchy which involves a nontrivial modification for the fermionic flows iii for the simplest constrained mr skp hierarchy skp12 12 we show that the orbit of darboux backlund transformations lies on a supersymmetric toda lattice being a square root of the standard one dimensional toda lattice and also we find explicit wronskian ratio solutions for the super tau function

canonical gauge equivalences of the sakns and stb hierarchies we study the gauge transformations between the supersymmetric akns sakns and supersymmetric two boson stb hierarchies the hamiltonian nature of these gauge transformations is investigated which turns out to be canonical we also obtain the darboux backlund transformations for the sakns hierarchy from these gauge transformations

convergence and loss bounds for bayesian sequence prediction the probability of observing x sub t at time t given past observations x sub 1 x sub t 1 can be computed if the true generating distribution spl mu of the sequences x sub 1 x sub 2 x sub 3 is known if spl mu is unknown but known to belong to a class spl mscr one can base one s prediction on the bayes mix spl xi defined as a weighted sum of distributions spl nu spl isin spl mscr various convergence results of the mixture posterior spl xi sub t to the true posterior spl mu sub t are presented in particular a new elementary derivation of the convergence spl xi sub t spl mu sub t spl rarr 1 is provided which additionally gives the rate of convergence a general sequence predictor is allowed to choose an action y sub t based on x sub 1 x sub t 1 and receives loss spl lscr sub x t y t if x sub t is the next symbol of the sequence no assumptions are made on the structure of spl lscr apart from being bounded and spl mscr the bayes optimal prediction scheme spl lambda sub spl xi based on mixture spl xi and the bayes optimal informed prediction scheme spl lambda sub spl mu are defined and the total loss l sub spl xi of spl lambda sub spl xi is bounded in terms of the total loss l sub spl mu of spl lambda sub spl mu it is shown that l sub spl xi is bounded for bounded l sub spl mu and l sub spl xi l sub spl mu spl rarr 1 for l sub spl mu spl rarr spl infin convergence of the instantaneous losses is also proven

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

hamiltonian structures of generalized manin radul super kdv and constrained super kp hierarchies a study of hamiltonian structures associated with supersymmetric lax operators is presented following a constructive approach the hamiltonian structures of inami kanno super kdv hierarchy and constrained modified super kp hierarchy are investigated from the reduced supersymmetric gelfand dickey brackets by applying a gauge transformation on the hamiltonian structures associated with these two nonstandard super lax hierarchies we obtain the hamiltonian structures of generalized manin radul super kdv and constrained super kp hierarchies we also work out a few examples and compare them with the known results

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

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

lie algebras and hamiltonian structures of multi component ablowitz kaup newell segur hierarchy isospectral and non isospectral hierarchies of multi component ablowitz kaup newell segur akns are obtained from a matrix spectral problem then by means of the zero curvature representations of the isospectral flows km and non isospectral flows n we construct the symmetries and their algebraic structures for isospectral multi component akns hierarchies demonstrate the recursive operator l is a strong and hereditary symmetry for the isospectral hierarchy we also derive that there are implectic operator and symplectic operator j such that l j and discuss the multi hamiltonian structures and the liouville integrability of the isospectral hierarchies

heisenberg subalgebra dressing approach and super bi hamiltonian integrable system starting with the affine super algebra osp 2 2 we have shown how different choices of heisenberg subalgebras in conjunction with the dressing operator approach of wilson 1958 leads to different super symmetric bi hamiltonian systems subsequently the explicit form of the symplectic operators are deduced with the help of the trace identity lastly a possible relation with the tau function is indicated

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

integrable rosochatius deformations of the restricted soliton flows a method to construct integrable rosochatius deformations of the restricted soliton flows in the setup of lax formulation is presented the integrable rosochatius deformations of the restricted soliton flows such as the restricted ablowitz kaup newell segur flow the restricted tu meng flow the restricted tu flow with neumann type constraints and the restricted modified korteweg de vries flow together with their lax representations are presented in addition a lax representation of the jacobi rosochatius system is obtained

lie algebras and korteweg de vries equations abstract for each metrizable lie algebra g an infinite commuting hierarchy of bi hamiltonian integrable extensions of the korteweg de vries kdv hierarchy is found the classical kdv hierarchy itself corresponds to the case g 0 the construction is based on the hamiltonian formalism on the dual space to a certain one parameter family of differential lie algebras produced from g

on the non hereditary recursion operator and the constraint on the potential associated with the giachetti johnson equation and its gauge equivalent yang equation the recursion operator associated with the giachetti johnson gj equation is shown not to be hereditary restricting the potential to the invariant subspace of the recursion operator leads to a constraint on the potential under the constraint two systems obtained from the gj equation and the related time evolution equation for eigenfunctions are shown to be naturally consistent constants of the motion for former system are given and a solution to this system satisfies a certain higher order stationary equation also similar results are obtained for the yang equation

soliton scattering with amplitude changes of a negative order akns equation the paper investigates a nonlinear model which provides soliton scattering with amplitude changes this is a nonlinear equation in negative order akns hierarchy we derive its n soliton solutions through hirota method for dynamics 1 soliton is characterized by an asymmetric shape and time dependent amplitude then by means of asymptotic analysis it is found that two soliton scattering provides not only phase shifts but also amplitude changes for each soliton which differs from classical soliton interactions where only phase shifts appear appendix a gives a tentative discussion on lax integrability and formal hamiltonian structures of the negative order hierarchy and appendix b lists reductions of equations and solutions

constraints of the 2 1 dimensional integrable soliton systems the authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2 1 dimensions can be constrained to integrable hierarchies in 1 1 dimensions such that submanifolds solutions of the given systems in 2 1 can be obtained by solving the resulting integrable systems in 1 1 dimensions the constraints of the kp hierarchy to the akns and burgers hierarchies respectively are shown in detail and the results of these for the modified kp and 2 1 dimensional analogue of the caudrey dodd gibbon kotera sawata equations to several integrable systems in 1 1 are given

a note on the super akns equations the authors find some relationships between the usual akns scheme with the super one when its elements take values from the grassman algebra on a two dimensional vector case the solutions of these super akns equations are discussed

hamiltonian structure of the super evolution equation the constrained variational calculus proposed in previous papers y zheng y li and d chen sci sinica a 24 138 1986 g tu kexue tangbao 29 1227 1984 is generalized to the supersymmetric case utilizing this method to some super akns system which has soliton solutions and conserved quantities their equations of motion with the hamiltonian structure can be written as a 4 4 matrix this is the supersymmetric symplectic matrix and hn are hamiltonians of the system the conserved quantities worked out in recent literature are just the first few terms of our analytic expression

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

integrable nonlinear evolution equations with time dependent coefficients a simple and straightforward method for generating completely integrable nonlinear evolution equations with time dependent coefficients is presented herein for the equations under consideration the solutions to those given by vector fields which are independent of time are given thus explicit links between equations are obtained as an application of the proposed method it is shown that the linear superposition with arbitrary time dependent coefficients of different members of an integrable hierarchy is again integrable furthermore it turns out that for some integrable equations like the korteweg de vries kdv the benjamin ono bo or the kadomtsev petviashvili kp the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow those flows demonstrated for the kdv which can be generated by lie homomorphisms coming from first order problems are completely classified many well known equations which can be found in the literature are of that type as an application of such a first order link a direct link is given from kdv to the cylindrical kdv and from there to the kp with nontrivial dependence on the second spatial variable

compound gauss markov random fields for image estimation algorithms for obtaining approximations to statistically optimal estimates for images modeled as compound gauss markov random fields are discussed the authors consider both the maximum a posteriori probability map estimate and the minimum mean squared error mmse estimate for image estimation and image restoration compound image models consist of several submodels having different characteristics along with an underlying structure model which govern transitions between these image submodels two different compound random field models are employed the doubly stochastic gaussian dsg random field and a compound gauss markov cgm random field the authors present map estimators for dsg and cgm random fields using simulated annealing a fast converging algorithm called deterministic relaxation which however converges to only a locally optimal map estimate is also presented as an alternative for reducing computational loading on sequential machines for comparison purposes the authors include results on the fixed lag smoothing mmse estimator for the dsg field and its suboptimal m algorithm approximation

a novel symmetry constraint of the super ckdv system a new 1 1 dimensional integrable system i e the super coupled korteweg de vries ckdv system has been constructed by a super extension of the well known 1 1 dimensional ckdv system for this new system a novel symmetry constraint between the potential and the eigenfunction can be obtained by means of the binary nonlinearization of its lax pairs the constraints for even variables are explicit and the constraints for odd variables are implicit under the symmetry constraint the spacial part and the temporal parts of the equations associated with the lax pairs for the super ckdv system can be decomposed into the super finite dimensional integrable hamiltonian systems on the supersymmetry manifold r4n 2n 2 whose integrals of motion are explicitly given

simulated annealing in compound gaussian random fields image processing recently a stochastic relaxation technique called simulated annealing has been developed to search for a globally optimal solution in image estimation and restoration problems the convergence of simulated annealing has been proved only for random fields with a compact range space because of this images were modeled as random fields with bounded discrete or continuous values however in most image processing problems it is more natural to model the image as a random field with values in a noncompact space e g conditional gaussian models the proof of convergence of the stochastic relaxation method is extended to a class of compound gauss markov random fields simulation results are provided to show the power of these methods

single image defogging by multiscale depth fusion restoration of fog images is important for the deweathering issue in computer vision the problem is ill posed and can be regularized within a bayesian context using a probabilistic fusion model this paper presents a multiscale depth fusion mdf method for defog from a single image a linear model representing the stochastic residual of nonlinear filtering is first proposed multiscale filtering results are probabilistically blended into a fused depth map based on the model the fusion is formulated as an energy minimization problem that incorporates spatial markov dependence an inhomogeneous laplacian markov random field for the multiscale fusion regularized with smoothing and edge preserving constraints is developed a nonconvex potential adaptive truncated laplacian is devised to account for spatially variant characteristics such as edge and depth discontinuity defog is solved by an alternate optimization algorithm searching for solutions of depth map by minimizing the nonconvex potential in the random field the mdf method is experimentally verified by real world fog images including cluttered depth scene that is challenging for defogging at finer details the fog free images are restored with improving contrast and vivid colors but without over saturation quantitative assessment of image quality is applied to compare various defog methods experimental results demonstrate that the accurate estimation of depth map by the proposed edge preserved multiscale fusion should recover high quality images with sharp details

integrable fermionic extensions of the garnier system and the anharmonic oscillator we first derive a fermionic extension of the garnier system by applying the method of binary nonlinearization of spectral problem to the supersymmetric kdv equation of kupershmidt and then a fermionic extension of the anharmonic oscillator by a simple reduction the integrable properties of resulting systems such as lax representations corresponding r matrices and conversed integrals of motion are established

the virasoro and kac moody symmetries for the principal chiral model for the principal chiral model two hierarchies of symmetries and the related virasoro and kac moody algebras are derived by using a recursion operator

the mean field theory in em procedures for blind markov random field image restoration a markov random field mrf model based em expectation maximization procedure for simultaneously estimating the degradation model and restoring the image is described the mrf is a coupled one which provides continuity inside regions of smooth gray tones and discontinuity at region boundaries constraints for the restoration problem which is in general ill posed the computational difficulty associated with the em procedure for mrfs is resolved by using the mean field theory from statistical mechanics an orthonormal blur decomposition is used to reduce the chances of undesirable locally optimal estimates experimental results on synthetic and real world images show that this approach provides good blur estimates and restored images the restored images are comparable to those obtained by a wiener filter in mean square error but are most visually pleasing

lie algebraic structure for the akns system a new approach is proposed to construct the master symmetry and to prove its lie algebraic structure for the akns system

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

symmetries and constants of motion for new akns hierarchies two sets of infinite number of symmetries their lie algebra properties and constants of motion for a class of non linear evolution equations associated with non isospectral deformations of the akns spectral problem are constructed

integrable non isospectral flows associated with the kadomtsev petviashvili equations in 2 1 dimensions the symmetries of the kadomtsev petviashvili kp equations in 2 1 dimensions yield two hierarchies of integrable non linear evolution equations nee one is the familiar family of isospectral flows the kp hierarchy the other is non isospectral and its flows have coefficients which depend linearly on x and y the spectral methods used to solve kp can be used to solve all these nee an underlying infinite dimensional lie algebra is used to determine all the lax pairs for both families and it also determines their symplectic structures constants of the motion are constructed for the non isospectral cases

parameter reduction for the compound gauss markov model the efficacy of the compound gauss markov cgm model initially proposed by jeng and woods 1990 has been demonstrated in several image processing applications however parameter estimation for the cgm model is difficult since it is not clear as to how the constraints or interdependence amongst the model parameters can be incorporated into the estimation procedures as result the parameter estimates tend to be inconsistent it is shown that under some reasonable symmetry constraints the 80 interdependent parameters of the cgm model can be reduced to seven independent ones this guarantees the consistency of model parameters obtained from parameter estimation algorithms thereby removing a main obstacle for the parameter estimation of the cgm model

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

algorithmic complexity bounds on future prediction errors we bound the future loss when predicting any computably stochastic sequence online solomonoff finitely bounded the total deviation of his universal predictor m from the true distribution m by the algorithmic complexity of m here we assume that we are at a time t 1 and have already observed x x 1 x t we bound the future prediction performance on x t 1x t 2 by a new variant of algorithmic complexity of m given x plus the complexity of the randomness deficiency of x the new complexity is monotone in its condition in the sense that this complexity can only decrease if the condition is prolonged we also briefly discuss potential generalizations to bayesian model classes and to classification problems

a family of liouville integrable lattice equations with a parameter and its two symmetry constraints a two by two matrix spectral problem with an arbitrary constant is introduced a family of integrable lattice equations involving a parameter is derived by the technique of lax pairs the typical lattice equation in the obtained family is a deformed toda lattice system the bi hamiltonian structure of the obtained family of integrable lattice equations is established by using the discrete trace identity and we prove the liouville integrability of obtained family of integrable lattice equations then bargmann symmetry constraint and an implicit symmetry constraint of the obtained family are presented by binary nonlinearization method under these two symmetry constraints we obtain two series of integrable symplectic maps and two sequences of completely integrable finite dimensional hamiltonian systems in liouville sense and every integrable lattice equation in the obtained family is decomposed by an integrable symplectic map and a completely integrable finite dimensional hamiltonian system as a specia

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

the fully supersymmetric akns equations the general fully supersymmetric approach to the akns method is presented the supersymmetric kdv and mkdv equations are considered as examples the supersymmetric backlund transformation is constructed for the supersymmetric kdv equation

on calibration error of randomized forecasting algorithms it has been recently shown that calibration with an error less than d 0 is almost surely guaranteed with a randomized forecasting algorithm where forecasts are obtained by random rounding the deterministic forecasts up to d we show that this error cannot be improved for a vast majority of sequences we prove that using a probabilistic algorithm we can effectively generate with probability close to one a sequence resistant to any randomized rounding forecasting with an error much smaller than d we also reformulate this result by means of a probabilistic game

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

remarks on the complete integrability of dynamical systems with fermionic variables the authors study the role of 1 1 graded tensor field t in the analysis of complete integrability of dynamical systems with fermionic variables they find that such a tensor t can be a recursion operator if and only if t is even as a graded map namely if and only if p t 0 they clarify this fact by constructing an odd tensor for two examples a supersymmetric toda chain and a supersymmetric harmonic oscillator they explicitly show that it cannot be a recursion operator since it does not allow new constants of motion to be built from the first two in contrast to what usually happens with ordinary i e nongraded systems

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

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

a 3 sx 3 matrix spectral problem for akns hierarchy and its binary nonlinearization abstract a 3 3 matrix spectral problem for akns soliton hierarchy is introduced and the corresponding bargmann symmetry constraint involving lax pairs and adjoint lax pairs is discussed an explicit new poisson algebra is proposed and thus the liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system the obtained nonlinearized lax systems in which the nonlinearized spatial system is intimately related to stationary akns flows lead to a sort of new involutive solutions to each akns soliton equation therefore the binary nonlinearization theory is successfully extended to a case of 3 3 matrix spectral problem for akns hierarchy

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

sato s backlund transformations additional symmetries and asvm formula for the discrete kp hierarchy two kinds of symmetries sato s backlund transformations and additional symmetries for the discrete kp dkp hierarchy are introduced and the asvm formula which demonstrates the equivalence of these two kinds of symmetries is obtained in this process the fay identity and the difference fay identity of the dkp hierarchy are introduced and the asvm formula in the form of tau function is calculated

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

optimality of universal bayesian sequence prediction for general loss and alphabet various optimality properties of universal sequence predictors based on bayes mixtures in general and solomonoff s prediction scheme in particular will be studied the probability of observing xt at time t given past observations x1 xt 1 can be computed with the chain rule if the true generating distribution of the sequences x1x2x3 is known if is unknown but known to belong to a countable or continuous class one can base ones prediction on the bayes mixture defined as a w weighted sum or integral of distributions the cumulative expected loss of the bayes optimal universal prediction scheme based on is shown to be close to the loss of the bayes optimal but infeasible prediction scheme based on we show that the bounds are tight and that no other predictor can lead to significantly smaller bounds furthermore for various performance measures we show pareto optimality of and give an occam s razor argument that the choice w sim 2 k for the weights is optimal where k is the length of the shortest program describing the results are applied to games of chance defined as a sequence of bets observations and rewards the prediction schemes and bounds are compared to the popular predictors based on expert advice extensions to infinite alphabets partial delayed and probabilistic prediction classification and more active systems are briefly discussed

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