multi lagrangians hereditary operators and lax pairs for the korteweg de vries positive and negative hierarchies we present an approach to the construction of action principles the inverse problem of the calculus of variations for first order in time derivatives differential equations and generalize it to field theory in order to construct systematically for integrable equations which are based on the existence of a nijenhuis or hereditary operator a multi lagrangian ladder of action principles which is complementary to the well known multi hamiltonian formulation we work out results for the korteweg de vries kdv equation which is a member of the positive hierarchy related to a hereditary operator three negative hierarchies of negative evolution equations are defined naturally from the hereditary operator as well in a concise way suitable for field theory the euler lagrange equations arising from the action principles are equivalent to deformations of the original evolution equation and the deformations are obtained explicitly in terms of the positive and negative evolution vectors we recognize after appropriate coordinate transformations the liouville sinh gordon hunter zheng and camassa holm equations as negative evolution equations the multi lagrangian ladder for kdv is directly mappable to a ladder for any of these negative equations and other positive evolution equations e g the harry dym and a special case of the krichever novikov equations for example several nonequivalent nonlocal time reparametrization invariant action principles for kdv are constructed and a new nonlocal action principle for the deformed system sinh gordon spatial translation vector is presented local and nonlocal hamiltonian operators are obtained in factorized form as the inverses of all the nonequivalent symplectic two forms in the ladder alternative lax pairs for all negative evolution vectors are constructed using the negative vectors and the hereditary operator as only input this result leads us to conclude that basically all positive and negative evolution equations in the hierarchies share the same infinite dimensional sets of local and nonlocal constants of the motion for kdv which are explicitly obtained using symmetries and the local and nonlocal action principles for kdv

a simplified analysis of plate structures using trefftz functions this paper presents a method for a quick evaluation of stresses and displacements in a plate structure for elastostatic problems first an approximation is built from the mathematical structure of the solutions for starshaped domains which is representative of long wavelength effects second a variational formulation consistent with this approximation is proposed finally two alternative procedures for coupling adjacent domains are demonstrated

information theoretic subspace clustering this paper addresses the problem of grouping the data points sampled from a union of multiple subspaces in the presence of outliers information theoretic objective functions are proposed to combine structured low rank representations lrrs to capture the global structure of data and information theoretic measures to handle outliers in theoretical part we point out that group sparsity induced measures ell 2 1 norm ell alpha norm and correntropy can be justified from the viewpoint of half quadratic hq optimization which facilitates both convergence study and algorithmic development in particular a general formulation is accordingly proposed to unify hq based group sparsity methods into a common framework in algorithmic part we develop information theoretic subspace clustering methods via correntropy with the help of parzen window estimation correntropy is used to handle either outliers under any distributions or sample specific errors in data pairwise link constraints are further treated as a prior structure of lrrs based on the hq framework iterative algorithms are developed to solve the nonconvex information theoretic loss functions experimental results on three benchmark databases show that our methods can further improve the robustness of lrr subspace clustering and outperform other state of the art subspace clustering methods

bayesian approaches to gaussian mixture modeling a bayesian based methodology is presented which automatically penalizes overcomplex models being fitted to unknown data we show that with a gaussian mixture model the approach is able to select an optimal number of components in the model and so partition data sets the performance of the bayesian method is compared to other methods of optimal model selection and found to give good results the methods are tested on synthetic and real data sets

dual hybrid trefftz element formulation based on independent boundary traction frame reference epfl article 102727view record in web of science record created on 2007 04 23 modified on 2016 08 08

lagrangian multiforms and multidimensional consistency we show that well chosen lagrangians for a class of two dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice on the basis of this property we formulate a lagrangian description for such systems in terms of lagrangian multiforms we discuss the connection of this formalism with the notion of multidimensional consistency and the role of the lattice from the point of view of the relevant variational principle

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

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

one class classifiers based on entropic spanning graphs one class classifiers offer valuable tools to assess the presence of outliers in data in this paper we propose a design methodology for one class classifiers based on entropic spanning graphs our approach also takes into account the possibility to process nonnumeric data by means of an embedding procedure the spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier the final partition is derived by exploiting a criterion based on mutual information minimization here we compute the mutual information by using a convenient formulation provided in terms of the alpha jensen difference once training is completed in order to associate a confidence level with the classifier decision a graph based fuzzy model is constructed the fuzzification process is based only on topological information of the vertices of the entropic spanning graph as such the proposed one class classifier is suitable also for data characterized by complex geometric structures we provide experiments on well known benchmarks containing both feature vectors and labeled graphs in addition we apply the method to the protein solubility recognition problem by considering several representations for the input samples experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state of the art approaches

covariant poisson brackets in geometric field theory we establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter as introduced by crnkovic witten and zuckerman from the multisymplectic form the main result is that the poisson bracket associated with this symplectic structure according to the standard rules is precisely the covariant bracket due to peierls and dewitt

stochastic blind equalization based on pdf fitting using parzen estimator this work presents a new blind equalization approach that aims to force the probability density function pdf at the equalizer output to match the known constellation pdf quadratic distance between pdf s is used as the cost function to be minimized the proposed method relies on the parzen window method to estimate the data pdf and is implemented by a stochastic gradient descent algorithm the kernel size of the parzen estimator allows a dual mode switch or a soft switch between blind and decision directed equalization the proposed method converges faster than the constant modulus algorithm cma working at the symbol rate with a similar computational burden and reduces the residual error of the cma in multilevel modulations at the same time a comparison with the most common blind techniques is presented

multisymplectic approach to integrable defects in the sine gordon model ideas from the theory of multisymplectic systems introduced recently in integrable systems by the author and kundu to discuss liouville integrability in classical field theories with a defect are applied to the sine gordon model the key ingredient is the introduction of a second poisson bracket in the theory that allows for a hamiltonian description of the model that is completely equivalent to the standard one in the absence of a defect in the presence of a defect described by frozen backlund transformations our approach based on the new bracket unifies the various tools used so far to attack the problem it also gets rid of the known issues related to the evaluation of the poisson brackets of the defect matrix which involve fields at coinciding space point the location of the defect the original lagrangian approach also finds a nice reinterpretation in terms of the canonical transformation representing the defect conditions

discrete pluriharmonic functions as solutions of linear pluri lagrangian systems pluri lagrangian systems are variational systems with the multi dimensional consistency property this notion has its roots in the theory of pluriharmonic functions in the z invariant models of statistical mechanics in the theory of variational symmetries going back to noether and in the theory of discrete integrable systems a d dimensional pluri lagrangian problem can be described as follows given a d form l on an m dimensional space m d whose coefficients depend on a function u of m independent variables called field find those fields u which deliver critical points to the action functionals s sigma int sigma l for any d dimensional manifold in the m dimensional space we investigate discrete 2 dimensional linear pluri lagrangian systems i e those with quadratic lagrangians l the action is a discrete analogue of the dirichlet energy and solutions are called discrete pluriharmonic functions we classify linear pluri lagrangian systems with lagrangians depending on diagonals they are described by generalizations of the star triangle map examples of more general quadratic lagrangians are also considered

numerical conditioning problems and solutions for nonparametric i i d statistical active contours in this paper we propose an active contour model based on nonparametric independent and identically distributed i i d statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background this is not however the first active contour model proposed to solve the segmentation problem under these same assumptions in contrast to prior active contour models based on nonparametric i i d statistics we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour instead treating the segmentation problem as a pixel wise classification problem we formulate an active contour to minimize the unbiased pixel wise average misclassification probability amp this not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour but it also solves a numerical conditioning problem that arises with such prior active contour models as a result the amp model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i i d statistical segmentation problem via probability distances to discuss this improved numerical behavior more precisely we introduce the notion of x201c conditioning ratio x201d and demonstrate that the proposed amp active contour is numerically better conditioned i e exhibits a much smaller conditioning ratio than prior probability distance based active contours

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

multi time lagrangian 1 forms for families of backlund transformations toda type systems general lagrangian theory of discrete one dimensional integrable systems is illustrated by a detailed study of backlund transformations for toda type systems commutativity of backlund transformations is shown to be equivalent to the consistency of the system of discrete multi time euler lagrange equations the precise meaning of the commutativity in the periodic case when all maps are double valued is established it is shown that the gluing of different branches is governed by the so called superposition formulas the closure relation for the multi time lagrangian 1 form on solutions of the variational equations is proved for all toda type systems superposition formulas are instrumental for this proof the closure relation was previously shown to be equivalent to the spectrality property of backlund transformations i e to the fact that the derivative of the lagrangian with respect to the spectral parameter is a common integral of motion of the family of backlund transformations we relate this integral of motion to the monodromy matrix of the zero curvature representation which is derived directly from equations of motion in an algorithmic way this serves as further evidence in favor of the idea that backlund transformations serve as zero curvature representations for themselves

a robust information clustering algorithm we focus on the scenario of robust information clustering ric based on the minimax optimization of mutual information mi the minimization of mi leads to the standard mass constrained deterministic annealing clustering which is an empirical risk minimization algorithm the maximization of mi works out an upper bound of the empirical risk via the identification of outliers noisy data points furthermore we estimate the real risk vc bound and determine an optimal cluster number of the ric based on the structural risk minimization principle one of the main advantages of the minimax optimization of mi is that it is a nonparametric approach which identifies the outliers through the robust density estimate and forms a simple data clustering algorithm based on the square error of the euclidean distance

a nonparametric clustering algorithm with a quantile based likelihood estimator clustering is a representative of unsupervised learning and one of the important approaches in exploratory data analysis by its very nature clustering without strong assumption on data distribution is desirable information theoretic clustering is a class of clustering methods that optimize information theoretic quantities such as entropy and mutual information these quantities can be estimated in a nonparametric manner and information theoretic clustering algorithms are capable of capturing various intrinsic data structures it is also possible to estimate information theoretic quantities using a data set with sampling weight for each datum assuming the data set is sampled from a certain cluster and assigning different sampling weights depending on the clusters the cluster conditional information theoretic quantities are estimated in this letter a simple iterative clustering algorithm is proposed based on a nonparametric estimator of the log likelihood for weighted data sets the clustering algorithm is also derived from the principle of conditional entropy minimization with maximum entropy regularization the proposed algorithm does not contain a tuning parameter the algorithm is experimentally shown to be comparable to or outperform conventional nonparametric clustering methods

a new cluster isolation criterion based on dissimilarity increments this paper addresses the problem of cluster defining criteria by proposing a model based characterization of interpattern relationships taking a dissimilarity matrix between patterns as the basic measure for extracting group structure dissimilarity increments between neighboring patterns within a cluster are analyzed empirical evidence suggests modeling the statistical distribution of these increments by an exponential density we propose to use this statistical model which characterizes context to derive a new cluster isolation criterion the integration of this criterion in a hierarchical agglomerative clustering framework produces a partitioning of the data while exhibiting data interrelationships in terms of a dendrogram type graph the analysis of the criterion is undertaken through a set of examples showing the versatility of the method in identifying clusters with arbitrary shape and size the number of clusters is intrinsically found without requiring ad hoc specification of design parameters nor engaging in a computationally demanding optimization procedure

reduction groups and related integrable difference systems of nonlinear schrodinger type we extend the reduction group method to the lax darboux schemes associated with nonlinear schrodinger type equations we consider all possible finite reduction groups and construct corresponding lax operators darboux transformations hierarchies of integrable differential difference equations integrable partial difference systems and associated scalar partial difference equations

information cut for clustering using a gradient descent approach we introduce a new graph cut for clustering which we call the information cut it is derived using parzen windowing to estimate an information theoretic distance measure between probability density functions we propose to optimize the information cut using a gradient descent based approach our algorithm has several advantages compared to many other graph based methods in terms of determining an appropriate affinity measure computational complexity memory requirements and coping with different data scales we show that our method may produce clustering and image segmentation results comparable or better than the state of the art graph based methods

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

legclust a clustering algorithm based on layered entropic subgraphs hierarchical clustering is a stepwise clustering method usually based on proximity measures between objects or sets of objects from a given data set the most common proximity measures are distance measures the derived proximity matrices can be used to build graphs which provide the basic structure for some clustering methods we present here a new proximity matrix based on an entropic measure and also a clustering algorithm legciust that builds layers of subgraphs based on this matrix and uses them and a hierarchical agglomerative clustering technique to form the clusters our approach capitalizes on both a graph structure and a hierarchical construction moreover by using entropy as a proximity measure we are able with no assumption about the cluster shapes to capture the local structure of the data forcing the clustering method to reflect this structure we present several experiments on artificial and real data sets that provide evidence on the superior performance of this new algorithm when compared with competing ones

convergence of gradient descent for minimum error entropy principle in linear regression we study the convergence of minimum error entropy mee algorithms when they are implemented by gradient descent this method has been used in practical applications for more than one decade but there has been no consistency or rigorous error analysis this paper gives the first rigorous proof for the convergence of the gradient descent method for mee in a linear regression setting the mean square error is proved to decay exponentially fast in terms of the iteration steps and of order o frac 1 m in terms of the sample size m the mean square convergence is guaranteed when the step size is chosen appropriately and the scaling parameter is large enough

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

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

symmetries and integrability of discrete equations defined on a black white lattice we study the deformations of the h equations presented recently by adler bobenko and suris which are naturally defined on a black white lattice for each one of these equations two different three leg forms are constructed leading to two different discrete toda type equations their multidimensional consistency leads to backlund transformations relating different members of this class as well as to lax pairs their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries

vorticity and symplecticity in multi symplectic lagrangian gas dynamics the lagrangian multi dimensional ideal compressible gas dynamic equations are written in a multi symplectic form in which the lagrangian fluid labels m i the lagrangian mass coordinates and time t are the independent variables and in which the eulerian position of the fluid element and the entropy are the dependent variables constraints in the variational principle are incorporated by means of lagrange multipliers the constraints are the entropy advection equation s t 0 the lagrangian map equation where is the fluid velocity and the mass continuity equation which has the form where is the jacobian of the lagrangian map in which and is the specific volume of the gas the internal energy per unit volume of the gas corresponds to a non barotropic gas the lagrangian is used to define multi momenta and to develop de donder weyl hamiltonian equations the de donder weyl equations are cast in a multi symplectic form the pullback conservation laws and the symplecticity conservation laws are obtained one class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws and another class of symplecticity laws are related to derivatives of the lagrangian energy conservation law with respect to the lagrangian mass coordinates m i we show that the vorticity symplecticity laws can be derived by a lie dragging method and also by using noether s second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action we obtain the cartan poincare form describing the equations and we discuss a set of differential forms representing the equation system

on covariant poisson brackets in classical field theory how to give a natural geometric definition of a covariant poisson bracket in classical field theory has for a long time been an open problem as testified by the extensive literature on multisymplectic poisson brackets together with the fact that all these proposals suffer from serious defects on the other hand the functional approach does provide a good candidate which has come to be known as the peierls de witt bracket and whose construction in a geometrical setting is now well understood here we show how the basic multisymplectic poisson bracket already proposed in the 1970s can be derived from the peierls de witt bracket applied to a special class of functionals this relation allows to trace back most if not all of the problems encountered in the past to ambiguities the relation between differential forms on multiphase space and the functionals they define is not one to one and also to the fact that this class of functionals does not form a poisson subalgebra

resolution of three dimensional stokes fluid flows using a trefftz method a new method to solve three dimensional linear stokes problems is proposed in this paper it is shown that in using extended trefftz polynomials associated with an original weak formulation which is suited to these special functions it is possible to get a high precision at a reasonnable numerical coast three dimensional examples show the accuracy of the proposed approach which is then applied to domain optimization as for the free surface problem

multi time lagrangian 1 forms for families of backlund transformations relativistic toda type systems we establish the pluri lagrangian structure for families of backlund transformations of relativistic toda type systems the key idea is a novel embedding of these discrete time one dimensional systems into certain two dimensional 2d pluri lagrangian lattice systems this embedding allows us to identify the corner equations which are the main building blocks of the multi time euler lagrange equations with local superposition formulae for backlund transformations these superposition formulae in turn are key ingredients necessary to understand and to prove commutativity of the multi valued backlund transformations furthermore we discover a 2d generalization of the spectrality property known for families of backlund transformations this result produces a family of local conservations laws for 2d pluri lagrangian lattice systems with densities being derivatives of the discrete 2 form with respect to the backlund spectral parameter thus a relation of the pluri lagrangian structure with more traditional integrability notions is established

lagrangian multiform structure for the lattice kp system we present a lagrangian for the bilinear discrete kp or hirota miwa equation furthermore we show that this lagrangian can be extended to a lagrangian 3 form when embedded in a higher dimensional lattice obeying a closure relation thus we establish the multiform structure as proposed in lobb and nijhoff 2009 j phys a math theor 42 454013 in a higher dimensional case

the hybrid trefftz finite element model and its application to plate bending this paper presents a new hybrid element approach and applies it to plate bending in contrast to more conventional models the formulation is based on displacement fields which fulfil a priori the non homogeneous lagrange equation trefftz method the interelement continuity is enforced by using a stationary principle together with an independent interelement displacement the final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs the formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve sided elements to be generated where relevant known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain for example particular singular corner elements elements presenting circular holes etc thus a high degree of accuracy may be achieved without a troublesome mesh refinement another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements triangles quadrilaterals etc presenting an increasing degree of accuracy the paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements the conclusions present some ideas concerning the adaptive version of the new elements extension to nonlinear problems and some other developments

exactly solved models on planar graphs with vertices in z 3 it is shown how exactly solved edge interaction models on the square lattice may be extended onto more general planar graphs with edges connecting a subset of next nearest neighbour vertices of mathbb z 3 this is done by using local deformations of the square lattice that arise through the use of the star triangle relation similar to baxter s z invariance property these local deformations leave the partition function invariant up to some simple factors coming from the star triangle relation the deformations used here extend the usual formulation of z invariance by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model the quasi classical limit is also considered in which case the deformations imply a classical z invariance property as well as a related local closure relation for the action functional of a system of classical discrete laplace equations

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

combining multiple clusterings using evidence accumulation we explore the idea of evidence accumulation eac for combining the results of multiple clusterings first a clustering ensemble a set of object partitions is produced given a data set n objects or patterns in d dimensions different ways of producing data partitions are 1 applying different clustering algorithms and 2 applying the same clustering algorithm with different values of parameters or initializations further combinations of different data representations feature spaces and clustering algorithms can also provide a multitude of significantly different data partitionings we propose a simple framework for extracting a consistent clustering given the various partitions in a clustering ensemble according to the eac concept each partition is viewed as an independent evidence of data organization individual data partitions being combined based on a voting mechanism to generate a new n times n similarity matrix between the n patterns the final data partition of the n patterns is obtained by applying a hierarchical agglomerative clustering algorithm on this matrix we have developed a theoretical framework for the analysis of the proposed clustering combination strategy and its evaluation based on the concept of mutual information between data partitions stability of the results is evaluated using bootstrapping techniques a detailed discussion of an evidence accumulation based clustering algorithm using a split and merge strategy based on the k means clustering algorithm is presented experimental results of the proposed method on several synthetic and real data sets are compared with other combination strategies and with individual clustering results produced by well known clustering algorithms

a redundancy based measure of dissimilarity among probability distributions for hierarchical clustering criteria we introduce novel dissimilarity into a probabilistic clustering task to properly measure dissimilarity among multiple clusters when each cluster is characterized by a subpopulation in the mixture model this measure of dissimilarity is called redundancy based dissimilarity among probability distributions from aspects of both source coding and a statistical hypothesis test we shed light on several of the theoretical reasons for the redundancy based dissimilarity among probability distributions being a reasonable measure of dissimilarity among clusters we also elucidate a principle in common for the measures of redundancy based dissimilarity and ward s method in terms of hierarchical clustering criteria moreover we show several related theorems that are significant for clustering tasks in the experiments properties of the measure of redundancy based dissimilarity are examined in comparison with several other measures

multi lagrangians for integrable systems we propose a general scheme to construct multiple lagrangians for completely integrable nonlinear evolution equations that admit multi hamiltonian structure the recursion operator plays a fundamental role in this construction we use a conserved quantity higher lower than the hamiltonian in the potential part of the new lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map this leads to some remarkable new developments we show that nonlinear evolutionary systems that admit n fold first order local hamiltonian structure can be cast into variational form with 2n 1 lagrangians which will be local functionals of clebsch potentials this number increases to 3n 2 when the miura transformation is invertible furthermore we construct a new lagrangian for polytropic gas dynamics in 1 1 dimensions which is a free local functional of the physical field variables namely density and velocity thus dispensing with the necessity of introducing clebsch potentials entirely this is a consequence of bi hamiltonian structure with a compatible pair of first and third order hamiltonian operators derived from sheftel s recursion operator

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

lagrangian multiform structure for the lattice gel fand dikii hierarchy the lattice gel fand dikii hierarchy was introduced by nijhoff et al 1992 inverse problems 8 597 621 as the family of partial difference equations generalizing to higher rank the lattice korteweg de vries systems and includes in particular the lattice boussinesq system we present a lagrangian for the generic member of the lattice gel fand dikii hierarchy and show that it can be considered as a lagrangian 2 form when embedded in a higher dimensional lattice obeying a closure relation thus the multiform structure proposed by lobb and nijhoff 2009 j phys a math theor 42 454013 is extended to a multi component system

kernel risk sensitive loss definition properties and application to robust adaptive filtering nonlinear similarity measures defined in kernel space such as correntropy can extract higher order statistics of data and offer potentially significant performance improvement over their linear counterparts especially in non gaussian signal processing and machine learning in this paper we propose a new similarity measure in kernel space called the kernel risk sensitive loss krsl and provide some important properties we apply the krsl to adaptive filtering and investigate the robustness and then develop the mkrsl algorithm and analyze the mean square convergence performance compared with correntropy the krsl can offer a more efficient performance surface thereby enabling a gradient based method to achieve faster convergence speed and higher accuracy while still maintaining the robustness to outliers theoretical analysis results and superior performance of the new algorithm are confirmed by simulation

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

a trefftz approach to computational mechanics this paper presents a trefftz method for solving structural elasticity problems and flow problems of incompressible viscous fluids the problem of unilateral contact is also dealt with for each type of problem trefftz polynomials and associated variational formulations are given complex structures are studied by a sub structuring technique this method requires the resolution of a non symmetrical linear system it is shown that it is possible to take advantage of this trefftz approximation in two ways i the approach presented can be considered as a simplified method which enables a solution to be evaluated quickly ii this approach also makes it possible to obtain a good quality solution associated with high degree polynomial bases this method is adapted to optimization processes because the discretization of the structure requires only very few sub domains to build a good approximation and offers a great flexibility in use copyright 2003 john wiley sons ltd

insights into the robustness of minimum error entropy estimation the minimum error entropy mee is an important and highly effective optimization criterion in information theoretic learning itl for regression problems mee aims at minimizing the entropy of the prediction error such that the estimated model preserves the information of the data generating system as much as possible in many real world applications the mee estimator can outperform significantly the well known minimum mean square error mmse estimator and show strong robustness to noises especially when data are contaminated by non gaussian multimodal heavy tailed discrete valued and so on noises in this brief we present some theoretical results on the robustness of mee for a one parameter linear errors in variables eiv model and under some conditions we derive a region that contains the mee solution which suggests that the mee estimate can be very close to the true value of the unknown parameter even in presence of arbitrarily large outliers in both input and output variables theoretical prediction is verified by an illustrative example

determining the number of clusters using information entropy for mixed data in cluster analysis one of the most challenging and difficult problems is the determination of the number of clusters in a data set which is a basic input parameter for most clustering algorithms to solve this problem many algorithms have been proposed for either numerical or categorical data sets however these algorithms are not very effective for a mixed data set containing both numerical attributes and categorical attributes to overcome this deficiency a generalized mechanism is presented in this paper by integrating renyi entropy and complement entropy together the mechanism is able to uniformly characterize within cluster entropy and between cluster entropy and to identify the worst cluster in a mixed data set in order to evaluate the clustering results for mixed data an effective cluster validity index is also defined in this paper furthermore by introducing a new dissimilarity measure into the k prototypes algorithm we develop an algorithm to determine the number of clusters in a mixed data set the performance of the algorithm has been studied on several synthetic and real world data sets the comparisons with other clustering algorithms show that the proposed algorithm is more effective in detecting the optimal number of clusters and generates better clustering results

discrete time calogero moser system and lagrangian 1 form structure we study the lagrange formalism of the rational calogero moser cm system both in discrete time and continuous time as a first example of a lagrangian 1 form structure in the sense of the recent paper lobb and nijhoff 2009 j phys a math theor 42 454013 the discrete time model of the cm system was established some time ago arising as a pole reduction of a semi discrete version of the kadomtsev petviashvili kp equation and was shown to lead to an exactly integrable correspondence multivalued map in this paper we present the full kp solution based on the commutativity of the discrete time flows in the two discrete kp variables the compatibility of the corresponding lax matrices is shown to lead directly to the relevant closure relation on the level of the lagrangians performing successive continuum limits on both the level of the kp equation and the level of the cm system we establish the proper lagrangian 1 form structure for the continuum case of the cm model we use the example of the three particle case to elucidate the implementation of the novel least action principle which was presented in lobb and nijhoff 2009 for the simpler case of lagrangian 1 forms

a reproducing kernel hilbert space framework for information theoretic learning this paper provides a functional analysis perspective of information theoretic learning itl by defining bottom up a reproducing kernel hilbert space rkhs uniquely determined by the symmetric nonnegative definite kernel function known as the cross information potential cip the cip as an integral of the product of two probability density functions characterizes similarity between two stochastic functions we prove the existence of a one to one congruence mapping between the itl rkhs and the hilbert space spanned by square integrable probability density functions therefore all the statistical descriptors in the original information theoretic learning formulation can be rewritten as algebraic computations on deterministic functional vectors in the itl rkhs instead of limiting the functional view to the estimators as is commonly done in kernel methods a connection between the itl rkhs and kernel approaches interested in quantifying the statistics of the projected data is also established

continuous symmetric reductions of the adler bobenko suris equations continuously symmetric solutions of the adler bobenko suris class of discrete integrable equations are presented initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations these solutions are shown to be determined by an integrable system of partial differential equations the connection of this system to the nijhoff hone joshi generating partial differential equations is established and an auto backlund transformation and a lax pair for it are constructed applied to the h1 and q1 0 members of the adler bobenko suris family the method of continuously symmetric reductions yields explicit solutions determined by the painleve trancendents

generalized hyperbolic ernst equations for an einstein maxwell weyl field a new family of nonlinear partial differential equations is presented they represent a generalization of the hyperbolic ernst equations for an einstein mawxell weyl field in general relativity a backlund transformation for the system of equations under consideration is given and their direct relation to the complete boussinesq hierarchy of soliton equations is illustrated

on the structure of vector quantizers vector quantization is intrinsically superior to predictive coding transform coding and other suboptimal and em ad hoc procedures since it achieves optimal rate distortion performance subject only to a constraint on memory or block length of the observable signal segment being encoded the key limitation of existing techniques is the very large randomly generated code books which must be stored and the computational complexity of the associated encoding procedures the quantization operation is decomposed into its rudimentary structural components this leads to a simple and elegant approach to derive analytical properties of optimal quantizers some useful properties of quantizers and algorithmic approaches are given which are relevant to the complexity of both storage and processing in the encoding operation highly disordered quantizers which have been designed using a clustering algorithm are considered finally lattice quantizers are examined which circumvent the need for a code book by using a highly structured code based on lattices the code vectors are algorithmically generated in a simple manner rather than stored in a code book and fast algorithms perform the encoding algorithm with negligible complexity

the mee principle in data classification a perceptron based analysis this letter focuses on the issue of whether risk functionals derived from information theoretic principles such as shannon or renyi s entropies are able to cope with the data classification problem in both the sense of attaining the risk functional minimum and implying the minimum probability of error allowed by the family of functions implemented by the classifier here denoted by min pe the analysis of this so called minimization of error entropy mee principle is carried out in a single perceptron with continuous activation functions yielding continuous error distributions in spite of the fact that the analysis is restricted to single perceptrons it reveals a large spectrum of behaviors that mee can be expected to exhibit in both theory and practice in what concerns the theoretical mee our study clarifies the role of the parameters controlling the perceptron activation function of the squashing type in often reaching the minimum probability of error our study also clarifies the role of the kernel density estimator of the error density in achieving the minimum probability of error in practice

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

a super integrable discretization of the calogero model a time discretization that preserves the super integrability of the calogero model is obtained by application of the integrable time discretization of the harmonic oscillator to the projection method for the calogero model with continuous time in particular the difference equations of motion which provide an explicit scheme for time integration are explicitly presented for the two body case numerical results exhibit that the scheme conserves all the 3 conserved quantities of the two body calogero model with a precision of the machine epsilon times the number of iterations

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

hamiltonian multiform description of an integrable hierarchy motivated by the notion of lagrangian multiforms which provide a lagrangian formulation of integrability and by the results of the authors on the role of covariant hamiltonian formalism for integrable field theories we propose the notion of hamiltonian multiforms for integrable 1 1 dimensional field theories they provide the hamiltonian counterpart of lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy for a given hierarchy taking a lagrangian multiform as the starting point we provide a systematic construction of a hamiltonian multiform based on a generalization of techniques of covariant hamiltonian field theory this also produces two other important objects a symplectic multiform and the related multi time poisson bracket they reduce to a multisymplectic form and the related covariant poisson bracket if we restrict our attention to a single flow in the hierarchy our framework offers an alternative approach to define and derive conservation laws for a hierarchy we illustrate our results on three examples the potential korteweg de vries hierarchy the sine gordon hierarchy in light cone coordinates and the ablowitz kaup newell segur hierarchy

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

analytically tractable case of fuzzy c means clustering in this paper we offer a simple and accurate clustering algorithm which was derived as a closed form analytical solution to a cluster fit function minimization problem as a result the algorithm finds the global minimum of the fit function and combines exceptional efficiency with optimal clustering results

on the lagrangian structure of 3d consistent systems of asymmetric quad equations recently the first named author gave a classification of 3d consistent 6 tuples of quad equations with the tetrahedron property several novel asymmetric 6 tuples have been found due to 3d consistency these 6 tuples can be extended to discrete integrable systems on we establish lagrangian structures and closeness of discrete lagrangian 2 forms for the class of discrete integrable systems involving equations for which some but not all of the biquadratics are non degenerate this class covers among others some of the above mentioned novel systems

discrete time toda systems in this paper we discuss several concepts of the modern theory of discrete integrable systems including time discretization based on the notion of backlund transformation symplectic realizations of multi hamiltonian structures interrelations between discrete 1d systems and lattice 2d systems multi dimensional consistency as integrability of discrete systems interrelations between integrable systems of quad equations and integrable systems of laplace type pluri lagrangian structure as integrability of discrete variational systems all these concepts are illustrated by the discrete time toda lattices and their relativistic analogs

on backlund transformations for many body systems using the n particle periodic toda lattice and the relativistic generalization due to ruijsenaars of the elliptic calogero moser system as examples we revise the basic properties of the backlund transformations bts from the hamiltonian point of view the analogy between the bt and baxter s quantum q operator pointed out by pasquier and gaudin is exploited to produce a conjugated variable for the parameter of the bt such that belongs to the spectrum of the lax operator as a consequence the generating function of the composition of n bts gives rise to another canonical transformation separating variables for the model for the toda lattice the dual bt parametrized by is introduced

extended z invariance for integrable vector and face models and multi component integrable quad equations in a previous paper kels in j phys a 50 49 495202 2017 the author has established an extension of the z invariance property for integrable edge interaction models of statistical mechanics that satisfy the star triangle relation str form of the yang baxter equation ybe in the present paper an analogous extended z invariance property is shown to also hold for integrable vector models and interaction round a face irf models of statistical mechanics respectively as for the previous case of the str the z invariance property is shown through the use of local cubic type deformations of a 2 dimensional surface associated to the models which allow an extension of the models onto a subset of next nearest neighbour vertices of mathbb z 3 while leaving the partition functions invariant these deformations are permitted as a consequence of the respective ybe s satisfied by the models the quasi classical limit is also considered and it is shown that an analogous z invariance property holds for the variational formulation of classical discrete laplace equations which arise in this limit from this limit new integrable 3d consistent multi component quad equations are proposed which are constructed from a degeneration of the equations of motion for irf boltzmann weights

a family of integrable nonlinear equations of hyperbolic type a new system of integrable nonlinear equations of hyperbolic type obtained by a two dimensional reduction of the anti self dual yang mills equations is presented it represents a generalization of the ernst weyl equation of general relativity related to colliding neutrino and gravitational waves as well as of the fourth order equation of schwarzian type related to the kdv hierarchies which was introduced by nijhoff hone and joshi recently an auto backlund transformation of the new system is constructed leading to a superposition principle remarkably similar to the one connecting four solutions of the kdv equation at the level of the ernst weyl equation this backlund transformation and the associated superposition principle yield directly a generalization of the single and double harrison transformations of the ernst equation respectively the very method of construction also allows for revealing in an essentially algorithmic fashion other integrability features of the main subsystems such as their reduction to the painleve transcendents

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

liouville arnold integrability of the pentagram map on closed polygons the pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane this map has recently attracted a considerable interest mostly because its connection to a number of different domains such as classical projective geometry algebraic combinatorics moduli spaces cluster algebras and integrable systems integrability of the pentagram map was conjectured by schwartz and proved by the present authors for a larger space of twisted polygons in this article we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons in the case of convex polygons in the real projective plane this result implies the existence of a toric foliation on the moduli space the leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic our proof is based on an invariant poisson structure on the space of twisted polygons we prove that the hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants

information theoretic clustering clustering is an important topic in pattern recognition since only the structure of the data dictates the grouping unsupervised learning information theory is an obvious criteria to establish the clustering rule the paper describes a novel valley seeking clustering algorithm using an information theoretic measure to estimate the cost of partitioning the data set the information theoretic criteria developed here evolved from a renyi entropy estimator a renyi 1960 that was proposed recently and has been successfully applied to other machine learning applications j c principe et al 2000 an improved version of the k change algorithm is used in optimization because of the stepwise nature of the cost function and existence of local minima even when applied to nonlinearly separable data the new algorithm performs well and was able to find nonlinear boundaries between clusters the algorithm is also applied to the segmentation of magnetic resonance imaging data mri with very promising results

quantized minimum error entropy criterion comparing with traditional learning criteria such as mean square error the minimum error entropy mee criterion is superior in nonlinear and non gaussian signal processing and machine learning the argument of the logarithm in renyi s entropy estimator called information potential ip is a popular mee cost in information theoretic learning the computational complexity of ip is however quadratic in terms of sample number due to double summation this creates the computational bottlenecks especially for large scale data sets to address this problem in this paper we propose an efficient quantization approach to reduce the computational burden of ip which decreases the complexity from o n 2 to o mn with m ll n the new learning criterion is called the quantized mee qmee some basic properties of qmee are presented illustrative examples with linear in parameter models are provided to verify the excellent performance of qmee

hamiltonian multivector fields and poisson forms in multisymplectic field theory we present a general classification of hamiltonian multivector fields and of poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories this is a prerequisite for computing explicit expressions for the poisson bracket between two poisson forms