bihamiltonian structure of the two component kadomtsev petviashvili hierarchy of type b we employ a lax pair representation of the two component kadomtsev petviashvili hierarchy of type b and construct its bihamiltonian structure with r matrix techniques

specificity a graph based estimator of divergence in statistical modeling there are various techniques used to build models from training data quantitative comparison of modeling techniques requires a method for evaluating the quality of the fit between the model probability density function pdf and the training data one graph based measure that has been used for this purpose is the specificity we consider the large numbers limit of the specificity and derive expressions which show that it can be considered as an estimator of the divergence between the unknown pdf from which the training data was drawn and the model pdf built from the training data experiments using artificial data enable us to show that these limiting large number relations enable us to obtain good quantitative and qualitative predictions of the behavior of the measured specificity even for small numbers of training examples and in some extreme cases we demonstrate that specificity can provide a more sensitive measure of difference between various modeling methods than some previous graph based techniques key points are illustrated using real data sets we thus establish a proper theoretical basis for the previously ad hoc concept of specificity and obtain useful insights into the application of specificity in the analysis of real data

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

the r matrix theory and the reduction of poisson manifolds the relation between the r matrix approach to the construction of bi hamiltonian structures and the reduction of poisson manifolds is discussed the crucial role of the reduction process is emphasized by showing on the basis of specific examples that the knowledge of a r matrix may not be sufficient the reduction of the poisson structures related with r from the entire algebra to the lax submanifold may become an obstruction in constructing the bi hamiltonian structures of a specific hierarchy of integrable equations

peakons r matrix and toda lattice abstract the integrability of a family of hamiltonian systems describing in a particular case the motion of n peakons special solutions of the so called camassa holm equation is established in the framework of the r matrix approach starting from its lax representation in the general case the r matrix is a dynamical one and has an interesting though complicated structure however for a particular choice of the relevant parameters in the hamiltonian the one corresponding to the pure peakons case the r matrix becomes essentially constant and reduces to the one pertaining to the finite non periodic toda lattice intriguing consequences of this property are discussed and an integrable time discretisation is derived

mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

parameter space warping shape based correspondence between morphologically different objects this paper presents a novel and comprehensive method for the automated determination of correspondences between two morphologically different two dimensional 2 d or three dimensional 3 d objects correspondences are determined by warping parametric representations of the objects to be matched the warp is guided by the minimization of a similarity criterion function that measures features related to structural correspondence including euclidian point to point distance and differences in normals and curvature the method uses a continuous harmonic parameterization for both the object and the warp which provides 1 a high degree of computational efficiency 2 robust extraction of differential features not subject to discretization errors or noise amplification in differentiation 3 direct formulation of constraints to avoid overlaps in the resulting correspondence set and 4 a scale space paradigm of object shape and warp the new method does not search for individual landmarks but operates with a complete integrated representation of the object geometry the method was tested on 2 d and 3 d objects with substantial shape differences results demonstrated substantial improvements of 2 33 in correspondence accuracy and 15 59 in correspondence quality compared with direct registration methods

shape analysis of elastic curves in euclidean spaces this paper introduces a square root velocity srv representation for analyzing shapes of curves in euclidean spaces under an elastic metric in this srv representation the elastic metric simplifies to the il2 metric the reparameterization group acts by isometries and the space of unit length curves becomes the unit sphere the shape space of closed curves is the quotient space of a submanifold of the unit sphere modulo rotation and reparameterization groups and we find geodesics in that space using a path straightening approach these geodesics and geodesic distances provide a framework for optimally matching deforming and comparing shapes these ideas are demonstrated using 1 shape analysis of cylindrical helices for studying protein structure 2 shape analysis of facial curves for recognizing faces 3 a wrapped probability distribution for capturing shapes of planar closed curves and 4 parallel transport of deformations for predicting shapes from novel poses

poisson structures of calogero moser and ruijsenaars schneider models we examine the hamiltonian structures of some calogero moser and ruijsenaars schneider n body integrable models we propose explicit formulations of the bi hamiltonian structures for the discrete models and field theoretical realizations of these structures we discuss the relevance of these realizations as collective field theory for the discrete models

building 3 d statistical shape models by direct optimization statistical shape models are powerful tools for image interpretation and shape analysis a simple yet effective way of building such models is to capture the statistics of sampled point coordinates over a training set of example shapes however a major drawback of this approach is the need to establish a correspondence across the training set in 2 d a correspondence is often defined using a set of manually placed landmarks and linear interpolation to sample the shape in between such annotation is however time consuming and subjective particularly when extended to 3 d in this paper we show that it is possible to establish a dense correspondence across the whole training set automatically by treating correspondence as an optimization problem the objective function we use for the optimization is based on the minimum description length principle which we argue is a criterion that leads to models with good compactness specificity and generalization ability we manipulate correspondence by reparameterizing each training shape we describe an explicit representation of reparameterization for surfaces in 3 d that makes it impossible to generate an illegal i e not one to one correspondence we also describe several large scale optimization strategies for model building and perform a detailed analysis of each approach finally we derive quantitative measures of model quality allowing meaningful comparison between models built using different methods results are given for several different training sets of 3 d shapes which show that the minimum description length models perform significantly better than other approaches

on deformations of standard r matrices for integrable infinite dimensional systems simple deformations with a parameter of classical r matrices which follow from decomposition of appropriate lie algebras are considered as a result nonstandard lax representations for some well known integrable systems are presented and integrable evolution equations are constructed

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

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

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

improvement of the k means clustering filtering algorithm in this paper we present a modified filtering algorithm mfa by making use of center variations to speed up clustering process our method first divides clusters into static and active groups we use the information of cluster displacements to reject unlikely cluster centers for all nodes in the kd tree we reduce the computational complexity of filtering algorithm fa through finding candidates for each node mainly from the set of active cluster centers two conditions for determining the set of candidate cluster centers for each node from active clusters are developed our approach is different from the major available algorithm which passes no information from one stage of iteration to the next theoretical analysis shows that our method can reduce the computational complexity in terms of the number of distance calculations of fa at each stage of iteration by a factor of fc ac where fc and ac are the numbers of total clusters and active clusters respectively compared with the fa our algorithm can effectively reduce the computing time and number of distance calculations it is noted that our proposed algorithm can generate the same clusters as that produced by hard k means clustering the superiority of our method is more remarkable when a larger data set with higher dimension is used

integrable systems and loop coproducts we present a generalization of a framework for the construction of classical integrable systems that we call loop coproduct formulation musso 2010 j phys a math theor 43 434026 in this paper the loop coproduct formulation includes systems of gelfand tsetlin type the linear r matrix formulation the sklyanin algebras the reflection algebras the coalgebra symmetry approach and some of its generalizations as particular cases showing that all these apparently different approaches have a common algebraic origin on the other hand all these subcases do not exhaust the domain of applicability of this new technique so that new possible directions of investigation do naturally emerge in this framework

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

an agglomerative clustering algorithm using a dynamic k nearest neighbor list in this paper a new algorithm is developed to reduce the computational complexity of ward s method the proposed approach uses a dynamic k nearest neighbor list to avoid the determination of a cluster s nearest neighbor at some steps of the cluster merge double linked algorithm dla can significantly reduce the computing time of the fast pairwise nearest neighbor fpnn algorithm by obtaining an approximate solution of hierarchical agglomerative clustering in this paper we propose a method to resolve the problem of a non optimal solution for dla while keeping the corresponding advantage of low computational complexity the computational complexity of the proposed method dknna fs dynamic k nearest neighbor algorithm with a fast search in terms of the number of distance calculations is o n 2 where n is the number of data points compared to fpnn with a fast search fpnn fs the proposed method using the same fast search algorithm dknna fs can reduce the computing time by a factor of 1 90 2 18 for the data set from a real image in comparison with fpnn fs dknna fs can reduce the computing time by a factor of 1 92 2 02 using the data set generated from three images compared to dla with a fast search dla fs dknna fs can decrease the average mean square error by 1 26 for the same data set

a fast vq codebook generation algorithm using codeword displacement in this paper we present a fast codebook generation algorithm called cgaucd codebook generation algorithm using codeword displacement by making use of the codeword displacement between successive partition processes by implementing a fast search algorithm named mfaupi modified fast algorithm using projection and inequality for vq encoding in the partition step of cgaucd the codebook generation time can be further reduced significantly using mfaupi the computing time of cgaucd can be reduced by a factor of 4 7 7 6 compared to generalized lloyd algorithm gla our proposed method can reduce the codebook generation time by a factor of 35 9 121 2 compared to the best codebook generation algorithm to our knowledge our approach can further reduce the corresponding computing time by 26 0 32 8 it is noted that our proposed algorithm can generate the same codebook as that produced by the gla the superiority of our method is more remarkable when a larger codebook is generated

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

continuous medial representation for anatomical structures the m rep approach pioneered by pizer 2003 is a powerful morphological tool that makes it possible to employ features derived from medial loci skeletons in shape analysis this paper extends the medial representation paradigm into the continuous realm modeling skeletons and boundaries of three dimensional objects as continuous parametric manifolds while also maintaining the proper geometric relationship between these manifolds the parametric representation of the boundary medial relationship makes it possible to fit shape based coordinate systems to the interiors of objects providing a framework for combined statistical analysis of shape and appearance our approach leverages the idea of inverse skeletonization where the skeleton of an object is defined first and the object s boundary is derived analytically from the skeleton this paper derives a set of sufficient conditions ensuring that inverse skeletonization is well posed for single manifold skeletons and formulates a partial differential equation whose solutions satisfy the sufficient conditions an efficient variational algorithm for deformable template modeling using the continuous medial representation is described and used to fit a template to the hippocampus in 87 subjects from a schizophrenia study with sub voxel accuracy and 95 mean overlap

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

using circulant symmetry to model featureless objects summary grenander miller 1994 describe a model for representing amorphous twodimensional objects with no obvious landmark each object is represented by n vertices around its perimeter and is described by deforming an n sided regular polygon using edge transformations a multivariate normal distribution with a block circulant covariance matrix is used to model these edge transformations the purpose of this paper is to describe in detail the statistical properties of this multivariate model and the eigenstructure of the covariance matrix various special cases of the model are considered including articulated models and conditional markov random field models we consider maximum likelihood based inference and the model is applied to some datasets to explore shape variability

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

separation of variables for quadratic algebras algebras of maillet and reflection equation algebras we consider the problem of variable separation for the classical integrable hamiltonian systems possessing gl n valued lax matrices satisfying quadratic poisson brackets of freidel and maillet phys lett b 262 2 3 278 1991 with spectral parameter dependent a b c d tensors we formulate in terms of the corresponding a b c d tensors a sufficient condition that guarantees that the separating functions a u b u of sklyanin commun math phys 150 1 181 1992 scott e print arxiv hep th 940303 1994 and gekhtman commun math phys 167 593 1995 produce canonical coordinates we consider an important subcase of a b c d algebras namely the case of classical reflection equation algebras e sklyanin j phys a math gen 21 10 2357 1988 and formulate in terms of the corresponding r s matrices the analogous sufficient condition that guarantees the canonicity of the constructed coordinates for the case s 0 we recover our previous results on the variable separation for quadratic sklyanin brackets b dubrovin and t skrypnyk j math phys 60 093506 2019 we consider an example of gl n gl n valued trigonometric a b c d tensors that satisfy the considered condition and find a class of lax matrices for them for which the obtained set of canonical coordinates is complete

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

a fast exact gla based on code vector activity detection this paper introduces a new method for reducing the number of distance calculations in the generalized lloyd algorithm gla which is a widely used method to construct a codebook in vector quantization reduced comparison search detects the activity of the code vectors and utilizes it on the classification of the training vectors for training vectors whose current code vector has not been modified we calculate distances only to the active code vectors a large proportion of the distance calculations can be omitted without sacrificing the optimality of the partition the new method is included in several fast gla variants reducing their running times over 50 on average

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

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

a two component generalization of the degasperis procesi equation we present two different hamiltonian extensions of the degasperis procesi equation to two component equations the construction is based on the observation that the second hamiltonian operator of the degasperis procesi equation could be considered as the dirac reduced poisson tensor of the second hamiltonian operator of the boussinesq equation the first extension is generated by the hamiltonian operator which is a dirac reduced operator of the generalized but degenerated second hamiltonian operator of the boussinesq equation the second one is obtained by the n 2 supersymmetric extension of the aforementioned method as the byproduct of this procedure we obtain the hamiltonian system of interacting equations which contains the camassa holm and degasperis procesi equations

backlund transformations for the kp and mkp hierarchies with self consistent sources using gauge transformations for the corresponding generating pseudo differential operators ln in terms of eigenfunctions and adjoint eigenfunctions we construct several types of auto backlund transformations for the kp hierarchy with self consistent sources kphscs and mkp hierarchy with self consistent sources mkphscs respectively the backlund transformations from kphscs to mkphscs are also constructed in this way

bi hamiltonian structures for integrable systems on regular time scales a construction of the bi hamiltonian structures for integrable systems on regular time scales is presented the trace functional on an algebra of pseudodifferential operators valid on an arbitrary regular time scale is introduced the linear poisson tensors and the related hamiltonians are derived the quadratic poisson tensors are given by the use of the recursion operators of the lax hierarchies the theory is illustrated by differential counterparts of ablowitz kaup newell segur and kaup broer hierarchies

a fast k means clustering algorithm using cluster center displacement in this paper we present a fast k means clustering algorithm fkmcucd using the displacements of cluster centers to reject unlikely candidates for a data point the computing time of our proposed algorithm increases linearly with the data dimension d whereas the computational complexity of major available kd tree based algorithms increases exponentially with the value of d theoretical analysis shows that our method can reduce the computational complexity of full search by a factor of sf and sf is independent of vector dimension the experimental results show that compared to full search our proposed method can reduce computational complexity by a factor of 1 37 4 39 using the data set from six real images compared with the filtering algorithm which is among the available best algorithms of k means clustering our algorithm can effectively reduce the computing time it is noted that our proposed algorithm can generate the same clusters as that produced by hard k means clustering the superiority of our method is more remarkable when a larger data set with higher dimension is used

diffeomorphic sulcal shape analysis on the cortex we present a diffeomorphic approach for constructing intrinsic shape atlases of sulci on the human cortex sulci are represented as square root velocity functions of continuous open curves in r 3 and their shapes are studied as functional representations of an infinite dimensional sphere this spherical manifold has some advantageous properties it is equipped with a riemannian l 2 metric on the tangent space and facilitates computational analyses and correspondences between sulcal shapes sulcal shape mapping is achieved by computing geodesics in the quotient space of shapes modulo scales translations rigid rotations and reparameterizations the resulting sulcal shape atlas preserves important local geometry inherently present in the sample population the sulcal shape atlas is integrated in a cortical registration framework and exhibits better geometric matching compared to the conventional euclidean method we demonstrate experimental results for sulcal shape mapping cortical surface registration and sulcal classification for two different surface extraction protocols for separate subject populations

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

faster and more robust point symmetry based k means algorithm based on the recently published point symmetry distance psd measure this paper presents a novel psd measure namely symmetry similarity level ssl operator for k means algorithm our proposed modified point symmetry based k means mpsk algorithm is more robust than the previous psk algorithm by su and chou not only the proposed mpsk algorithm is suitable for the symmetrical intra clusters as the psk algorithm does the proposed mpsk algorithm is also suitable for the symmetrical inter clusters in addition two speedup strategies are presented to reduce the time required in the proposed mpsk algorithm experimental results demonstrate the significant execution time improvement and the extension to the symmetrical inter clusters of the proposed mpsk algorithm when compared to the previous psk algorithm

continuous medial representation of brain structures using the biharmonic pde a new approach for constructing deformable continuous medial models for anatomical structures is presented medial models describe geometrical objects by first specifying the skeleton of the object and then deriving the boundary surface corresponding to the skeleton however an arbitrary specification of a skeleton will not be valid unless a certain set of sufficient conditions is satisfied the most challenging of these is the non linear equality constraint that must hold along the boundaries of the manifolds forming the skeleton the main contribution of this paper is to leverage the biharmonic partial differential equation as a mapping from a codimension 0 subset of euclidean space to the space of skeletons that satisfy the equality constraint the pde supports robust numerical solution on freeform triangular meshes providing additional flexibility for shape modeling the approach is evaluated by generating continuous medial models for a large dataset of hippocampus shapes generalizations to modeling more complex shapes and to representing branching skeletons are demonstrated

segmentation registration and measurement of shape variation via image object shape a model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives metrics are described that compute an object representation s prior probability of local geometry by reflecting variabilities in the net s node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation a paradigm for image analysis of deforming such a model to optimize a posteriori probability is described and this paradigm is shown to be usable as a uniform approach for object definition object based registration between images of the same or different imaging modalities and measurement of shape variation of an abnormal anatomical object compared with a normal anatomical object examples of applications of these methods in radiotherapy surgery and psychiatry are given

4 dimensional frobenius manifolds and painleve vi a frobenius manifold has tri hamiltonian structure if it is even dimensional and its spectrum is maximally degenerate we study the case of the lowest nontrivial dimension n 4 and show that under the assumption of semisimplicity the corresponding isomonodromic fuchsian system is described by the painleve hbox vi mu equation since the solutions of this equation are known to parametrize semisimple frobenius manifolds of dimension n 3 this leads to an explicit procedure mapping 3 dimensional frobenius structures of 4 dimensional ones and giving all tri hamiltonian structures in four dimensions we illustrate the construction by computing two examples in the framework of frobenius structures on hurwitz spaces

analysis of principal nested spheres a general framework for a novel non geodesic decomposition of high dimensional spheres or high dimensional shape spaces for planar landmarks is discussed the decomposition principal nested spheres leads to a sequence of submanifolds with decreasing intrinsic dimensions which can be interpreted as an analogue of principal component analysis in a number of real datasets an apparent one dimensional mode of variation curving through more than one geodesic component is captured in the one dimensional component of principal nested spheres while analysis of principal nested spheres provides an intuitive and flexible decomposition of the high dimensional sphere an interesting special case of the analysis results in finding principal geodesics similar to those from previous approaches to manifold principal component analysis an adaptation of our method to kendall s shape space is discussed and a computational algorithm for fitting principal nested spheres is proposed the result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation as exemplified by several datasets

local and non local multiplicative poisson vertex algebras and differential difference equations we develop the notions of multiplicative lie conformal and poisson vertex algebras local and non local and their connections to the theory of integrable differential difference hamiltonian equations we establish relations of these notions to q deformed w algebras and lattice poisson algebras we introduce the notion of adler type pseudodifference operators and apply them to integrability of differential difference hamiltonian equations

multi hamiltonian structures for a class of degenerate completely integrable systems in this paper a class of degenerate i e associated to a degenerate poisson structure completely integrable systems is studied which generalizes the so called odd and even master systems introduced and studied by mumford and by vanhaecke it is shown that all these completely integrable systems called the generalized master systems admit a multi hamiltonian formulation and a systematic construction of this multi hamiltonian structure is described

distance weighted discrimination high dimension low sample size statistical analysis is becoming increasingly important in a wide range of applied contexts in such situations the popular support vector machine suffers from data piling at the margin which can diminish generalizability this leads naturally to the development of distance weighted discrimination which is based on second order cone programming a modern computationally intensive optimization method

super toda lattices the lax formalism as described by oevel et al and in an earlier and more fundamental form by semenov kostant symes and adler can easily be generalized to the case where anticommuting variables are involved the so called supercase in this article this super lax formalism is applied to the well known associative superalgebra g mat m n subspaces of g to which the super poisson structures can be restricted arise in a natural way taking l in one of these subspaces formally leads to superextensions of the hierarchy of nonrelativistic toda lattices in the simplest case where only nearest neighbor interaction is involved the equations are explicitly solved furthermore the relevant two super hamiltonian structures are explicitly calculated finally a superextension of the relativistic toda lattice with a super hamiltonian structure is described herein

shape space smoothing splines for planar landmark data a method is developed for fitting smooth curves through a series of shapes of landmarks in two dimensions using unrolling and unwrapping procedures in riemannian manifolds an explicit method of calculation is given which is analogous to that of jupp kent 1987 for spherical data the resulting splines are called shape space smoothing splines the method resembles that of fitting smoothing splines in real spaces in that if the smoothing parameter is zero the resulting curve interpolates the data points and if it is infinitely large the curve is a geodesic line the fitted path to the data is defined such that its unrolled version at the tangent space of the starting point is a cubic spline fitted to the unwrapped data with respect to that path computation of the fitted path consists of an iterative procedure which converges quickly and the resulting path is given in a discretised form in terms of a piecewise geodesic path the procedure is applied to the analysis of some human movement data and a test for the appropriateness of a mean geodesic curve is given

lax nijenhuis operators for integrable systems the relationship between lax and bi hamiltonian formulations of dynamical systems on finite or infinite dimensional phase spaces is investigated the lax nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the lenard recursion relations while the converse holds for an operator with a simple spectrum explicit higher order hamiltonian structures for the toda system a second hamiltonian structure of the euler equation for a rigid body in n dimensional space and the quadratic adler gelfand dickey structure for the kdv hierarchy are derived using the lax nijenhuis equation

fast vector quantization algorithms based on nearest partition set search a fast search method for vector quantization is proposed in this paper it makes use of the fact that in the generalized lloyd algorithm gla a vector in a training sequence is either placed in the same minimum distance partition mdp as in the previous iteration or in a partition within a very small subset of partitions the proposed method searches for the mdp for a training vector only in this subset of partitions plus the single previous mdp as the size of this subset is much smaller than the total number of codevectors the search process is speeded up significantly the creation of the subset is essential as it has a direct effect on the improvement in computation time of the proposed method the schemes that create the subset efficiently have been proposed the proposed method generates a codebook identical to that generated using the gla it is simple and requires only minor modification of the gla and a modest amount of additional memory the experimental results show that the computation time of codebook training was improved by factors from 6 6 to 50 7 and from 5 8 to 70 4 for two test data sets when codebooks of sizes from n 16 to 2048 were trained the proposed method was also combined with an earlier published method to further improve the computation time

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

iterative shrinking method for clustering problems agglomerative clustering generates the partition hierarchically by a sequence of merge operations we propose an alternative to the merge based approach by removing the clusters iteratively one by one until the desired number of clusters is reached we apply local optimization strategy by always removing the cluster that increases the distortion the least data structures and their update strategies are considered the proposed algorithm is applied as a crossover method in a genetic algorithm and compared against the best existing clustering algorithms the proposed method provides best performance in terms of minimizing intra cluster variance

r structures yang baxter equations and related involution theorems there are several ways to construct functions in involution on a lie bi algebra a lie algebra equipped with a second lie bracket for the solvable systems associated to the casimir functions a second hamiltonian formulation can be constructed and a class of bi hamiltonian korteweg de vries like evolution equations with explicit space dependence is derived translating the casimir functions with the flow of a special vector field yields another set of functions in involution lenard relations are found for the corresponding hamiltonian systems finally solutions of the classical yang baxter equation lead to an analog of compatible hamiltonian pairs the invariants of the resulting hereditary operators are in involution

adaptive principal surfaces abstract we develop a nonlinear generalization of principal components analysis a principal surface of the data is constructed adaptively using some ideas from the mars procedure of friedman we explore applications to curve and surface reconstruction and to data summarization

tabu search algorithm for codebook generation in vector quantization abstract a tabu search algorithm is proposed for the codebook generation in vector quantization the key question is the definition of neighboring solution making random modifications to the current solution alone is not sufficient the proposed algorithm first makes non local changes to the codebook which is then fine tuned by the generalized lloyd algorithm gla for a set of gray scale images the new algorithm was better than gla alone and its results were comparable to simulated annealing for binary images the tabu search approach gave the best mse values

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

nonlocal quadratic poisson algebras monodromy map and bogoyavlensky lattices a new lax representation for the bogoyavlensky lattice is found and its r matrix interpretation is elaborated the r matrix structure turns out to be related to a highly nonlocal quadratic poisson structure on a direct sum of associative algebras the theory of such nonlocal structures is developed and the poisson property of the monodromy map is worked out in the most general situation some problems concerning the duality of lax representations are raised

what is the relativistic volterra lattice we develop a systematic procedure of finding integrable 6ldquo relativistic regular one parameter deformations for integrable lattice systems our procedure is based on the integrable time discretizations and consists of three steps first for a given system one finds a local discretization living in the same hierarchy second one considers this discretization as a particular cauchy problem for a certain 2 dimensional lattice equation and then looks for another meaningful cauchy problem which can be in turn interpreted as a new discrete time system third one has to identify integrable hierarchies to which these new discrete time systems belong these novel hierarchies are called then relativistic the small time step h playing the role of inverse speed of light we apply this procedure to the toda lattice and recover the well known relativistic toda lattice as well as to the volterra lattice and a certain bogoyavlensky lattice for which the relativistic deformations were not known previously

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

skeletal shape correspondence through entropy we present a novel approach for improving the shape statistics of medical image objects by generating correspondence of skeletal points each object s interior is modeled by an s rep i e by a sampled folded two sided skeletal sheet with spoke vectors proceeding from the skeletal sheet to the boundary the skeleton is divided into three parts the up side the down side and the fold curve the spokes on each part are treated separately and using spoke interpolation are shifted along that skeleton in each training sample so as to tighten the probability distribution on those spokes geometric properties while sampling the object interior regularly as with the surface boundary based correspondence method of cates et al entropy is used to measure both the probability distribution tightness and the sampling regularity here of the spokes geometric properties evaluation on synthetic and real world lateral ventricle and hippocampus data sets demonstrate improvement in the performance of statistics using the resulting probability distributions this improvement is greater than that achieved by an entropy based correspondence method on the boundary points

an improvement of the minimum distortion encoding algorithm for vector quantization in this note we present a very simple method for improving the efficiency of minimum distortion encoding for vector quantization simulations indicates a reduction of up to 70 percent in the number of multiplications for a full search vector quantizer with a large number of codewords and about 25 40 percent for a tree search vector quantizer similar improvement can be achieved in other vector quantization systems

elastic model based segmentation of 3 d neuroradiological data sets this paper presents a new technique for the automatic model based segmentation of three dimensional 3 d objects from volumetric image data the development closely follows the seminal work of taylor and cootes on active shape models but is based on a hierarchical parametric object description rather than a point distribution model the segmentation system includes both the building of statistical models and the automatic segmentation of new image data sets via a restricted elastic deformation of shape models geometric models are derived from a sample set of image data which have been segmented by experts the surfaces of these binary objects are converted into parametric surface representations which are normalized to get an invariant object centered coordinate system surface representations are expanded into series of spherical harmonics which provide parametric descriptions of object shapes it is shown that invariant object surface parametrization provides a good approximation to automatically determine object homology in terms of sets of corresponding sets of surface points gray level information near object boundaries is represented by 1 d intensity profiles normal to the surface considering automatic segmentation of brain structures as their driving application the authors choice of coordinates for object alignment was the well accepted stereotactic coordinate system major variation of object shapes around the mean shape also referred to as shape eigenmodes are calculated in shape parameter space rather than the feature space of point coordinates segmentation makes use of the object shape statistics by restricting possible elastic deformations into the range of the training shapes the mean shapes are initialized in a new data set by specifying the landmarks of the stereotactic coordinate system the model elastically deforms driven by the displacement forces across the object s surface which are generated by matching local intensity profiles elastical deformations are limited by setting bounds for the maximum variations in eigenmode space the technique has been applied to automatically segment left and right hippocampus thalamus putamen and globus pallidus from volumetric magnetic resonance scans taken from schizophrenia studies the results have been validated by comparison of automatic segmentation with the results obtained by interactive expert segmentation

fast agglomerative clustering using a k nearest neighbor graph we propose a fast agglomerative clustering method using an approximate nearest neighbor graph for reducing the number of distance calculations the time complexity of the algorithm is improved from o taun2 to o taun log n at the cost of a slight increase in distortion here tau denotes the lumber of nearest neighbor updates required at each iteration according to the experiments a relatively small neighborhood size is sufficient to maintain the quality close to that of the full search

a framework for automatic landmark identification using a new method of nonrigid correspondence a framework for automatic landmark identification is presented based on an algorithm for corresponding the boundaries of two shapes the auto landmarking framework employs a binary tree of corresponded pairs of shapes to generate landmarks automatically on each of a set of example shapes the landmarks are used to train statistical shape models known as point distribution models the correspondence algorithm locates a matching pair of sparse polygonal approximations one for each of a pair of boundaries by minimizing a cost function using a greedy algorithm the cost function expresses the dissimilarity in both the shape and representation error with respect to the defining boundary of the sparse polygons results are presented for three classes of shape which exhibit various types of nonrigid deformation

a novel approach for fast codebook re quantization in this paper we present a fast codebook re quantization algorithm fcra using codewords of a codebook being re quantized as the training vectors to generate the re quantized codebook our method is different from the available approach which uses the original training set to generate a re quantized codebook compared to the traditional approach our method can reduce the computing time dramatically since the number of codewords of a codebook being re quantized is usually much smaller than the number of original training vectors our method first classifies codewords of a re quantized codebook into static and active groups this approach uses the information of codeword displacements between successive partitions to reject impossible candidates in the partition process of codebook re quantization by implementing a fast search algorithm used for vector quantization encoding mfaupi in the partition step of fcra the computational complexity of codebook re quantization can be further reduced significantly using mfaupi the computing time of fcra can be reduced by a factor of 1 55 3 78 compared with the available approach oarc optimization algorithm for re quantization codebook our proposed method can reduce the codebook re quantization time by a factor of about 8005 using a training set of six real images this reduction factor is increased when the re quantized codebook size and or training set size are increased it is noted that our proposed algorithm can generate the same re quantized codebook as that produced by the oarc

a new vector quantization clustering algorithm the pairwise nearest neighbor pnn algorithm is presented as an alternative to the linde buzo gray 1980 lbg generalized lloyd 1982 algorithm for vector quantization clustering the pnn algorithm derives a vector quantization codebook in a diminishingly small fraction of the time previously required without sacrificing performance in addition the time needed to generate a codebook grows only o n log n in training set size and is independent of the number of code words desired using this method one can either minimize the number of code words needed subject to a maximum rate the pnn algorithm can be used with squared error and weighted squared error distortion measure simulations on a variety of images encoded at 1 2 b pixel indicate that pnn codebooks can be developed in roughly 5 of the time required by the lbg algorithm

the high dimension low sample size geometric representation holds under mild conditions summary high dimension low small sample size datasets have different geometrical properties from those of traditional low dimensional data in their asymptotic study regarding increasing dimensionality with a fixed sample size hall et al 2005 showed that each data vector is approximately located on the vertices of a regular simplex in a high dimensional space a perhaps unappealing aspect of their result is the underlying assumption which requires the variables viewed as a time series to be almost independent we establish an equivalent geometric representation under much milder conditions using asymptotic properties of sample covariance matrices we discuss implications of the results such as the use of principal component analysis in a high dimensional space extension to the case of nonindependent samples and also the binary classification problem

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes