symmetries and constants of the motion for higher order lagrangian systems we obtain a classification of infinitesimal symmetries of an autonomous higher order lagrangian system by using the theory of lifts of vector fields to tangent bundles also a classification of infinitesimal symmetries of a time dependent higher order lagrangian system is obtained the relationship between infinitesimal symmetries and constants of the motion is investigated

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

context aware sparse decomposition for image denoising and super resolution image prior models based on sparse and redundant representations are attracting more and more attention in the field of image restoration the conventional sparsity based methods enforce sparsity prior on small image patches independently unfortunately these works neglected the contextual information between sparse representations of neighboring image patches it limits the modeling capability of sparsity based image prior especially when the major structural information of the source image is lost in the following serious degradation process in this paper we utilize the contextual information of local patches denoted as context aware sparsity prior to enhance the performance of sparsity based restoration method in addition a unified framework based on the markov random fields model is proposed to tune the local prior into a global one to deal with arbitrary size images an iterative numerical solution is presented to solve the joint problem of model parameters estimation and sparse recovery finally the experimental results on image denoising and super resolution demonstrate the effectiveness and robustness of the proposed context aware method

n ary algebras a review with applications this paper reviews the properties and applications of certain n ary generalizations of lie algebras in a self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras particular attention will be paid to generalized lie algebras which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized jacobi identity and to filippov or n lie algebras which are defined by fully antisymmetric n brackets that satisfy the filippov identity three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this filippov algebras will be discussed at length including the cohomology complexes that govern their central extensions and their deformations it turns out that whitehead s lemma extends to all semisimple n lie algebras when the skewsymmetry of the lie or n lie algebra bracket is relaxed one is led to a more general type of n algebras the n leibniz algebras these will be discussed as well since they underlie the cohomological properties of n lie algebras the standard poisson structure may also be extended to the n ary case we shall review here the even generalized poisson structures whose generalized jacobi identity reproduces the pattern of the generalized lie algebras and the nambu poisson structures which satisfy the filippov identity and determine filippov algebras finally the recent work of bagger lambert and gustavsson on superconformal chern simons theory will be briefly discussed emphasis will be made on the appearance of the 3 lie algebra structure and on why the a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

geometric quantization of dirac manifolds we extend a prequantization procedure to dirac manifolds by using singular distributions obtained from 2 cocycles associated with dirac structures given a dirac manifold m d we describe a prequantization formula in terms of a lie algebroid connection and show that it is a representation of a poisson algebra consisting of admissible functions on m d on the space of global sections of a hermitian line bundle over m if and only if the curvature derived from the lie algebroid connection is represented by a skew symmetric operation which is naturally defined for m d moreover we describe a necessary and sufficient condition for the prequantization formula to be the representation in terms of a lie algebroid cohomology we introduce the notion of a polarization for m d and construct a representation of a subalgebra of admissible functions lastly we discuss procedures for quantization in two cases where m is compact and where m is not compact

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

connections between score matching contrastive divergence and pseudolikelihood for continuous valued variables score matching sm and contrastive divergence cd are two recently proposed methods for estimation of nonnormalized statistical methods without computation of the normalization constant partition function although they are based on very different approaches we show in this letter that they are equivalent in a special case in the limit of infinitesimal noise in a specific monte carlo method further we show how these methods can be interpreted as approximations of pseudolikelihood

lichnerowicz jacobi cohomology in this paper we extend the notion of lichnerowicz poisson cohomology to jacobi manifolds we study the relation of the so called lichnerowicz jacobi cohomology with the basic de rham cohomology and the cohomology of the lie algebra of functions relative to the representation defined by the hamiltonian vector fields a natural pairing with the canonical homology is constructed the relation between the lichnerowicz poisson cohomology of a quantizable poisson manifold and the lichnerowicz jacobi cohomology of the total space of a prequantization bundle is obtained particular cases of cosymplectic contact and locally conformal symplectic manifolds are discussed finally the lichnerowicz jacobi cohomology of a non transitive example is studied

the schouten nijenhuis bracket cohomology and generalized poisson structures newly introduced generalized poisson structures based on suitable skew symmetric contravariant tensors of even order are discussed in terms of the schouten nijenhuis bracket the associated jacobi identities are expressed as conditions on these tensors the cohomological contents of which is given in particular we determine the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

leibniz algebroid associated with a nambu poisson structure the notion of a leibniz algebroid is introduced and it is shown that each nambu poisson manifold has associated a canonical leibniz algebroid this fact permits one to define the modular class of a nambu poisson manifold as an appropriate cohomology class extending the well known modular class of poisson manifolds

multi peaked localized states of dnls in one and two dimensions abstract multi peaked localized stationary solutions of the discrete nonlinear schrodinger dnls equation are presented in one 1d and two 2d dimensions these are excited states of the discrete spectrum and correspond to multi breather solutions a simple very fast and efficient numerical method suggested by aubry has been used for their calculation the method involves no diagonalization but just iterations of a map starting from trivial solutions of the anti continuous limit approximate analytical expressions are presented and compared with the numerical results the linear stability of the calculated stationary states is discussed and the structure of the linear stability spectrum is analytically obtained for relatively large values of nonlinearity

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

co isotropic and legendre lagrangian submanifolds and conformal jacobi morphisms the notion of a co isotropic and legendre lagrangian submanifold of a jacobi manifold is given a characterization of conformal jacobi morphisms and conformal jacobi infinitesimal transformations is obtained as co isotropic and legendre lagrangian submanifolds of jacobi manifolds

reduction of jacobi manifolds a reduction procedure for jacobi manifolds is described in the algebraic setting of jacobi algebras as applications reduction by arbitrary submanifolds distributions and the reduction of jacobi manifolds with symmetry are discussed this generalized reduction procedure extends the well known reduction procedures for symplectic poisson contact and co symplectic structures

gradient vector fields on cosymplectic manifolds the authors study some geometrical properties of gradient vector fields on cosymplectic manifolds thereby emphasizing the close analogy with hamiltonian systems on symplectic manifolds it is shown that gradient vector fields and more generally local gradient vector fields can be characterized in terms of lagrangian submanifolds of the tangent bundle with respect to an induced symplectic structure in addition the symmetry and reduction properties of gradient vector fields are investigated

consistency of pseudolikelihood estimation of fully visible boltzmann machines a boltzmann machine is a classic model of neural computation and a number of methods have been proposed for its estimation most methods are plagued by either very slow convergence or asymptotic bias in the resulting estimates here we consider estimation in the basic case of fully visible boltzmann machines we show that the old principle of pseudolikelihood estimation provides an estimator that is computationally very simple yet statistically consistent

discrete breathers in nonlinear schrodinger hypercubic lattices with arbitrary power nonlinearity we study two specific features of onsite breathers in nonlinear schrodinger systems on d dimensional cubic lattices with arbitrary power nonlinearity i e arbitrary nonlinear exponent n their wavefunctions and energies close to the anti continuum limit small hopping limit and their excitation thresholds exact results are systematically compared to the predictions of the so called exponential ansatz ea and to the solution of the single nonlinear impurity model sni where all nonlinearities of the lattice but the central one where the breather is located have been removed in 1d the exponential ansatz is more accurate than the sni solution close to the anti continuum limit while the opposite result holds in higher dimensions the excitation thresholds predicted by the sni solution are in excellent agreement with the exact results but cannot be obtained analytically except in 1d an ea approach to the sni problem provides an approximate analytical solution that is asymptotically exact as n tends to infinity but the ea result degrades as the dimension d increases this is in contrast to the exact sni solution which improves as n and or d increase finally in our investigation of the sni problem we also prove a conjecture by bustamante and molina c a bustamante m i molina phys rev b 62 23 2000 15287 that the limiting value of the bound state energy is universal when n tends to infinity

geometry of four vector fields on quaternionic flag manifolds the purpose of this paper is to describe certain natural 4 vector fields on quaternionic flag manifolds which geometrically determine the bruhat cell decomposition these structures naturally descend from the symplectic group sp n and are related to the dressing action given by the iwasawa decomposition of the general linear group over the quaternions gln

estimation of non normalized statistical models by score matching one often wants to estimate statistical models where the probability density function is known only up to a multiplicative normalization constant typically one then has to resort to markov chain monte carlo methods or approximations of the normalization constant here we propose that such models can be estimated by minimizing the expected squared distance between the gradient of the log density given by the model and the gradient of the log density of the observed data while the estimation of the gradient of log density function is in principle a very difficult non parametric problem we prove a surprising result that gives a simple formula for this objective function the density function of the observed data does not appear in this formula which simplifies to a sample average of a sum of some derivatives of the log density given by the model the validity of the method is demonstrated on multivariate gaussian and independent component analysis models and by estimating an overcomplete filter set for natural image data

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

grammatical complexity for two dimensional maps we calculate the grammatical complexity of the symbol sequences generated from the henon map and the lozi map using the recently developed methods to construct the pruning front when the map is hyperbolic the language of symbol sequences is regular in the sense of the chomsky hierarchy and the corresponding grammatical complexity takes finite values it is found that the complexity exhibits a self similar structure as a function of the system parameter and the similarity of the pruning fronts is discussed as an origin of such self similarity for non hyperbolic cases it is observed that the complexity monotonically increases as we increase the resolution of the pruning front

dynamics of generalized poisson and nambu poisson brackets a unified setting for generalized poisson and nambu poisson brackets is discussed it is proved that a nambu poisson bracket of even order is a generalized poisson bracket characterizations of poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and lagrangian submanifolds of product and tangent manifolds respectively

the symplectic structure of euler lagrange superequations and batalin vilkoviski formalism we study the graded euler lagrange equations from the viewpoint of graded poincare cartan forms an application to a certain class of solutions of the batalin vilkoviski master equation is also given

homology and cohomology on generalized poisson manifolds the canonical homology of a generalized poisson manifold is introduced and the two spectral sequences associated with the periodic double complex are studied the generalized poisson cohomology is also discussed and a natural parity with the canonical homology is established the results hold for nambu poisson manifolds

parameter inference in a probabilistic model using clustered data abstract we propose a method to infer the parameters of a probabilistic model from given data samples our method is based on the pseudolikelihood and composite likelihood methods we cluster the given data samples and apply the clustered data samples to the pseudolikelihood and composite likelihood methods from an expansion of the pseudolikelihood method around the mean of a cluster the mean field and thouless anderson palmer equations are derived likewise from an expansion of the composite likelihood method around the mean of a cluster a method that is similar to the bethe approximation is derived we then perform numerical simulations using our method we find that our method gives an accurate estimate in the range of weak coupling parameters but has an inferior accuracy compared to the pseudolikelihood and composite likelihood methods in the range of strong coupling parameters in the range of strong coupling parameters as the number of clusters increases the inference accuracy of our method improves compared to the pseudolikelihood and composite likelihood methods our method reduces the number of computational tasks for the estimation therefore sacrificing the inference accuracy

nambu dirac manifolds we present a generalization of the dirac structure in the direction of the nambu poisson structure it is shown that with every integrable nambu dirac structure there is associated a leibniz algebroid which yields a singular foliation endowed with a closed form the examples of nambu dirac manifolds are dirac manifolds nambu poisson manifolds and manifolds with closed forms a different leibniz algebroid structure associated with a nambu poisson structure is adopted for this generalization

an algorithm to prune the area preserving henon map an explicit algorithm to provide the pruning front for the area preserving henon map is presented the procedure terminates within finitely many steps when the map has hyperbolic structure the only information required to specify the pruning front is a bifurcation diagram of homoclinic orbits and it is obtained by tracking orbits from the anti integrable limit the pruned region thus determined is used to construct the markov partition of the map and the topological entropy is evaluated as an application

spatial properties and numerical solitary waves of a nonintegrable discrete nonlinear schrdinger equation with nonlinear hopping in this paper we study a nonintegrable discrete nonlinear schrdinger dnls equation with nonlinear hopping by using the planar nonlinear dynamical map approach we address the spatial properties of the nonintegrable dnls equation through the constructions of exact period 1 and period 2 orbits of a planar nonlinear map which is a stationary version of the nonintegrable dnls equation we obtain the spatially periodic solutions of the nonintegrable dnls equation we also give the numerical simulations of the orbits of the planar nonlinear map and show how the nonlinear hopping terms affect those orbits by using discrete fourier transformation method we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dnls equation

the graded jacobi algebras and co homology jacobi algebroids i e jacobi versions of lie algebroids are studied in the context of graded jacobi brackets on graded commutative algebras this unifies various concepts of graded lie structures in geometry and physics a method of describing such structures by classical lie algebroids via certain gauging in the spirit of e witten s gauging of the exterior derivative is developed one constructs a corresponding cartan differential calculus graded commutative one in a natural manner this in turn gives canonical generating operators for triangular jacobi algebroids one gets in particular the lichnerowicz jacobi homology operators associated with classical jacobi structures courant jacobi brackets are obtained in a similar way and used to define an abstract notion of a courant jacobi algebroid and dirac jacobi structure

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

new generalized poisson structures new generalized poisson structures are introduced by using suitable skew symmetric contravariant tensors of even order the corresponding jacobi identities are provided by conditions on these tensors which may be understood as cocycle conditions as an example we provide the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

jacobi structures revisited jacobi algebroids i e graded lie brackets on the grassmann algebra associated with a vector bundle which satisfy a property similar to that of the jacobi brackets are introduced they turn out to be equivalent to generalized lie algebroids in the sense of iglesias and marrero jacobi bialgebroids are defined in the same manner a lifting procedure of elements of this grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding lie brackets is developed this gives the possibility of associating canonically a lie algebroid with any local lie algebra in the sense of kirillov

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

prequantizable poisson manifolds and jacobi structures we prove that the total space of a prequantization bundle of a prequantizable poisson manifold admits a jacobi structure and conversely a regular jacobi manifold is the prequantization bundle of a poisson manifold illustrative examples are presented

constrained time dependent lagrangian systems and lagrangian submanifolds in this paper a description a la tulczyjew of constrained time dependent dynamical systems in terms of special symplectic manifolds is first developed next this description is combined with the constraint algorithm for time dependent constrained systems

nambu tensors and commuting vector fields takhtajan has recently studied the consistency conditions for nambu brackets 1994 commun math phys 160 295 315 and suggested that they have to be skew symmetric and satisfy the leibnitz rule and the fundamental identity fi a generalization of the jacobi identity if the nth order nambu brackets in dimension n are written as where the summations range over the fi implies two conditions on the nambu tensor one algebraic and one differential the algebraic part of fi implies decomposability of and in this letter we show that the nambu bracket can then be written as where is the usual totally antisymmetric n dimensional tensor the summations range over and are n vector fields our main result is that the differential part of the fi is satisfied iff the vector fields commute examples are provided by integrable hamiltonian systems it turns out that then the nambu bracket itself guarantees that the motions stays on the manifold defined by the constants of motion of the integrable system while the n 1 nambu hamiltonians determine the possibly non integrable motion on this manifold

cosymplectic reduction for singular momentum maps in this paper we prove that the cosymplectic reduction of cosymplectic manifolds with symmetry due to c albert 1989 may be obtained from the marsden weinstein reduction theory we also study the reduction of cosymplectic manifolds with singular momentum map by using the results of sjamaar and lerman 1991 for the symplectic case

discrete compactons some exact results in this letter we use the method of constructing exact solutions on lattices proposed by kinnersley and described in schmidt 1979 phys rev b 20 4397 to obtain exact compacton solutions in discrete models we examine the linear stability of such solutions both for the bright compacton and for the dark compacton cases we focus on a quantization condition that the width of the profile should satisfy we also use this quantization condition to examine the possibility of compact coherent structures travelling in discrete settings our results are obtained for sinusoidal profiles and then generalized to elliptic functions of arbitrary modulus the possibility of multi compacton solutions is considered

nambu jacobi structures and jacobi algebroids we describe a nambu jacobi structure as a nambu poisson structure on a certain jacobi algebroid it is shown that the matched pair of leibniz algebroids for a nambu jacobi structure is the leibniz algebroid associated with this nambu poisson structure we also see that a different leibniz algebroid is associated with a nambu jacobi structure which is a natural generalization of the lie algebroid associated with a jacobi structure

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

generalized neighbor interaction models induced by nonlinear lattices it is shown that the tight binding approximation of the nonlinear schrodinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex both linear and nonlinear neighbor interactions the obtained lattices present nonstandard possibilities among which we mention a quasilinear regime where the pulse dynamics obeys essentially the linear schrodinger equation we analyze the properties of such models both in connection to their modulational stability as well as in regard to the existence and stability of their localized solitary wave solutions

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

a novel parameter estimation method for boltzmann machines we propose a novel estimator for a specific class of probabilistic models on discrete spaces such as the boltzmann machine the proposed estimator is derived from minimization of a convex risk function and can be constructed without calculating the normalization constant whose computational cost is exponential order we investigate statistical properties of the proposed estimator such as consistency and asymptotic normality in the framework of the estimating function small experiments show that the proposed estimator can attain comparable performance to the maximum likelihood expectation at a much lower computational cost and is applicable to high dimensional data

the z2 graded schouten nijenhuis bracket and generalized super poisson structures the super or z2 graded schouten nijenhuis bracket is introduced using it new generalized super poisson structures are found which are given in terms of certain graded skew symmetric contravariant tensors of even order the corresponding super jacobi identities are expressed by stating that these tensors have a zero super schouten nijenhuis bracket with themselves 0 as a particular case we provide the linear generalized super poisson structures which can be constructed on the dual spaces of simple superalgebras with a non degenerate killing metric the su 3 1 superalgebra is given as a representative example

comment on hamiltonian structures for the n dimensional lotka volterra equations plank s recent discussion in this journal of hamiltonian structure of lotka volterra dynamics is shown to have roots going back many years this is briefly sketched together with an infrastructure of the lie koenigs theorem and gibbs ensemble theory

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

on the geometric quantization of jacobi manifolds the geometric quantization of jacobi manifolds is discussed a natural cohomology termed lichnerowicz jacobi on a jacobi manifold is introduced and using it the existence of prequantization bundles is characterized to do this a notion of contravariant derivatives is used in such a way that the procedure developed by vaisman for poisson manifolds is naturally extended a notion of polarization is discussed and the quantization problem is studied the existence of prequantization representations is also considered

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

geometry of the dynamical systems with time dependent constraints and time dependent hamiltonian an approach towards quantization the quantization of a dynamical system with time dependent constraints or time dependent hamiltonians presents particular difficulties not only from a technical but also from a theoretical point of view we give here an approach to the evolution of such a quantum system in terms of star products on symplectic or canonical manifolds these star products are particular deformations of the usual product of functions deformations that start with the poisson bracket the introduction of the star products corresponds to a generalization of the weyl wigner quantization on a flat phase space this approach leads to the introduction of two times denoted t and playing different roles the time t plays the role of a geometrical time corresponding to the control while the other which parametrizes the evolution plays the role of a dynamical time the classical as well as the quantum results point out the role played by the variable t this reflects the coherence of the formalism

deformation quantization and nambu mechanics starting from deformation quantization star products the quantization problem of nambu mechanics is investigated after considering some impossibilities and pushing some analogies with field quantization a solution to the quantization problem is presented in the novel approach of zariski quantization of fields observables functions in this case polynomials this quantization is based on the factorization over of polynomials in several real variables we quantize the infinite dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is abelian associative and distributive this procedure is then adapted to derivatives needed for the nambu brackets which ensures the validity of the fundamental identity of nambu mechanics also at the quantum level our construction is in fact more general than the particular case considered here it can be utilized for quite general defining identities and for much more general star products

a block successive lower bound maximization algorithm for the maximum pseudo likelihood estimation of fully visible boltzmann machines maximum pseudo likelihood estimation mple is an attractive method for training fully visible boltzmann machines fvbms due to its computational scalability and the desirable statistical properties of the mple no published algorithms for mple have been proven to be convergent or monotonic in this note we present an algorithm for the mple of fvbms based on the block successive lower bound maximization bslm principle we show that the bslm algorithm monotonically increases the pseudo likelihood values and that the sequence of bslm estimates converges to the unique global maximizer of the pseudo likelihood function the relationship between the bslm algorithm and the gradient ascent ga algorithm for mple of fvbms is also discussed and a convergence criterion for the ga algorithm is given

training products of experts by minimizing contrastive divergence it is possible to combine multiple latent variable models of the same data by multiplying their probability distributions together and then renormalizing this way of combining individual expert models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data a product of experts poe is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary training a poe by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule fortunately a poe can be trained using a different objective function called contrastive divergence whose derivatives with regard to the parameters can be approximated accurately and efficiently examples are presented of contrastive divergence learning using several types of expert on several types of data

convergence of contrastive divergence algorithm in exponential family the contrastive divergence cd algorithm has achieved notable success in training energy based models including restricted boltzmann machines and played a key role in the emergence of deep learning the idea of this algorithm is to approximate the intractable term in the exact gradient of the log likelihood function by using short markov chain monte carlo mcmc runs the approximate gradient is computationally cheap but biased whether and why the cd algorithm provides an asymptotically consistent estimate are still open questions this paper studies the asymptotic properties of the cd algorithm in canonical exponential families which are special cases of the energy based model suppose the cd algorithm runs m mcmc transition steps at each iteration t and iteratively generates a sequence of parameter estimates theta t t ge 0 given an i i d data sample x i i 1 n sim p theta star under conditions which are commonly obeyed by the cd algorithm in practice we prove the existence of some bounded m such that any limit point of the time average sum s 0 t 1 theta s t as t to infty is a consistent estimate for the true parameter theta star our proof is based on the fact that theta t t ge 0 is a homogenous markov chain conditional on the data sample x i i 1 n this chain meets the foster lyapunov drift criterion and converges to a random walk around the maximum likelihood estimate the range of the random walk shrinks to zero at rate mathcal o 1 sqrt 3 n as the sample size n to infty

poisson jacobi reduction of homogeneous tensors the notion of homogeneous tensors is discussed we show that there is a one to one correspondence between multivector fields on a manifold m homogeneous with respect to a vector field on m and first order polydifferential operators on a closed submanifold n of codimension 1 such that is transversal to n this correspondence relates the schouten nijenhuis bracket of multivector fields on m to the schouten jacobi bracket of first order polydifferential operators on n and generalizes the poissonization of jacobi manifolds actually it can be viewed as a super poissonization this procedure of passing from a homogeneous multivector field to a first order polydifferential operator can also be understood as a sort of reduction in the standard case a half of a poisson reduction a dual version of the above correspondence yields in particular the correspondence between homogeneous symplectic structures on m and contact structures on n

dynamics of breathers in discrete nonlinear schro dinger models abstract we review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic non stationary excitations in discrete nonlinear schrodinger dnls models in two dimensions we show the existence of linearly stable stationary and non stationary localized vortex like solutions we also show that stationary on site localized excitations can have internal breathing modes which are spatially localized and symmetric the excitation of these modes leads to slowly decaying quasiperiodic oscillations finally we show that for some generalizations of the dnls equation where bistability occurs a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value

higher order simple lie algebras it is shown that the non trivial cocycles on simple lie algebras may be used to introduce antisymmetric multibrackets which lead to higher order lie algebras the definition of which is given their generalised jacobi identities turn out to be satisfied by the antisymmetric tensors or higher order structure constants which characterise the lie algebra cocycles this analysis allows us to present a classification of the higher order simple lie algebras as well as a constructive procedure for them our results are synthesised by the introduction of a single complete brst operator associated with each simple algebra

parameter inference in a probabilistic model from data regulation of transition rate in the monte carlo method abstract we consider the inference of parameters in a probabilistic model from a data set which is generated by an unknown probabilistic model the monte carlo method is a tool for obtaining a data set obeying a given probability distribution a set of transition rates is required to satisfy three conditions irreducible aperiodic and stationary for a sampled data set to represent a probability distribution we utilize the stationary condition of a probability distribution with respect to transition rates to infer parameters a frequency distribution by a data set substitutes for an unknown probability distribution in the condition our method includes minimum probability flow as a special case and becomes superior to it as the number of samples increases

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

on classification of intrinsic localized modes for the discrete nonlinear schrodinger equation abstract we consider localized modes discrete breathers of the discrete nonlinear schrodinger equation i d n dt n 1 n 1 2 n n 2 n 1 n z we study the diversity of the steady state solutions of the form n t ei tvn and the intervals of the frequency of their existence the base for the analysis is provided by the anticontinuous limit negative and large enough where all the solutions can be coded by the sequences of three symbols 0 and using dynamical systems approach we show that this coding is valid for 3 4533 and the point is a point of accumulation of saddle node bifurcations also we study other bifurcations of intrinsic localized modes which take place for and give the complete table of them for the solutions with codes consisting of less than four symbols

on one class of exact poisson structures we discuss some properties of a natural class of poisson structures on euclidean spaces and abstract manifolds in particular it is proved that such structures are always exact and may be reconstructed from their casimir functions it is shown that in low dimensions they give the whole class of exact poisson structures the dimension of poisson homology of these structures is computed in terms of the milnor number of their casimir functions we also analyze some concrete examples of such structures in low dimensions and show that their centers are generated by casimir functions

remarks on nambu poisson and nambu jacobi brackets we show that nambu poisson and nambu jacobi brackets can be defined inductively an n bracket n 2 is nambu poisson respectively nambu jacobi if and only if on fixing an argument we obtain an n 1 nambu poisson respectively nambu jacobi bracket as a by product we obtain relatively simple proofs of darboux type theorems for these structures

efficient learning in boltzmann machines using linear response theory the learning process in boltzmann machines is computationally very expensive the computational complexity of the exact algorithm is exponential in the number of neurons we present a new approximate learning algorithm for boltzmann machines based on mean field theory and the linear response theorem the computational complexity of the algorithm is cubic in the number of neurons in the absence of hidden units we show how the weights can be directly computed from the fixed point equation of the learning rules thus in this case we do not need to use a gradient descent procedure for the learning process we show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role finally we apply the method to a pattern completion task and show good performance for networks up to 100 neurons

on the geometric quantization of poisson manifolds in a paper by huebschmann j reine angew math 408 57 1990 the geometric quantization of poisson manifolds appears as a particular case of the quantization of poisson algebras here this quantization is presented straightforwardly the results include a geometric prequantization integrality condition and its discussion in particular cases such as dirac brackets an adaptation of the notion of a polarization and a construction of a quantum hilbert space and a computational example in the last section of the paper the general prequantization representations in the sense of urwin adv math 50 126 1983 are described for the poisson and jacobi manifolds

jacobi structures in r3 the most general jacobi brackets in r3 are constructed after solving the equations imposed by the jacobi identity two classes of jacobi brackets were identified according to the rank of the jacobi structures the associated hamiltonian vector fields are also constructed

on the higher order generalizations of poisson structures the characterization of the nambu poisson n tensors as a subfamily of the generalized poisson ones recently introduced and here extended to the odd order case is discussed the homology and cohomology complexes of both structures are compared and some physical considerations are made

consistent inference of a general model using the pseudolikelihood method recently a maximum pseudolikelihood mpl inference method has been successfully applied to statistical physics models with intractable likelihoods we use information theory to derive a relation between the pseudolikelihood and likelihood functions furthermore we show the consistency of the pseudolikelihood method for a general model

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

asymptotic normality of the maximum pseudolikelihood estimator for fully visible boltzmann machines boltzmann machines bms are a class of binary neural networks for which there have been numerous proposed methods of estimation recently it has been shown that in the fully visible case of the bm the method of maximum pseudolikelihood estimation mple results in parameter estimates which are consistent in the probabilistic sense in this brief we investigate the properties of mple for the fully visible bms further and prove that mple also yields an asymptotically normal parameter estimator these results can be used to construct confidence intervals and to test statistical hypotheses these constructions provide a closed form alternative to the current methods that require monte carlo simulation or resampling we support our theoretical results by showing that the estimator behaves as expected in simulation studies

generalized jacobi structures jacobi brackets a generalization of standard poisson brackets in which leibniz s rule is replaced by a weaker condition are extended to brackets involving an arbitrary even number of functions this new structure includes as a particular case the recently introduced generalized poisson structures the linear case on simple group manifolds is also studied and non trivial examples different from those coming from generalized poisson structures of this new construction are found by using the cohomology ring of the given group

solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity we study the existence and stability of localized states in the discrete nonlinear schrodinger equation on two dimensional nonsquare lattices the model includes both the nearest neighbor and long range interactions for the fundamental strongly localized soliton the results depend on the coordination number i e on the particular type of lattice the long range interactions additionally destabilize the discrete soliton or make it more stable if the sign of the interaction is respectively the same as or opposite to the sign of the short range interaction we also explore more complicated solutions such as twisted localized modes and solutions carrying multiple topological charge vortices that are specific to the triangular and honeycomb lattices in the cases when such vortices are unstable direct simulations demonstrate that they typically turn into zero vorticity fundamental solitons

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived