transformation groups for soliton equations iv a new hierarchy of soliton equations of kp type abstract a new approach to soliton equations based on functions or hirota s dependent variables vertex operators and the clifford algebra of free fermions is applied to study a new hierarchy of kadomtsev petviashvili type equations the bkp hierarchy the infinite dimensional orthogonal group acts on the space of bkp functions the sawada kotera equation is obtained as a reduction of bkp its infinitesimal transformations constitute the euclidean lie algebra a2 2

the adler shiota van moerbeke formula for the bkp hierarchy we study the bkp hierarchy and prove the existence of an adler shiota van moerbeke formula this formula relates the action of the bw1 algebra on tau functions to the action of the additional symmetries on wave functions

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

some results on the caudrey dodd gibbon kotera sawada equation in this paper a hierarchy of bilinear caudrey dodd gibbon kotera sawada equations with a unified structure is proposed a nonlinear superposition formula for the cdgks equation is proved under certain conditions a backlund transformation for a higher order cdgks equation is presented

the convergence of v cycle multigrid algorithms for axisymmetric laplace and maxwell equations we investigate some simple finite element discretizations for the axisymmetric laplace equation and the azimuthal component of the axisymmetric maxwell equations as well as multigrid algorithms for these discretizations our analysis is targeted at simple model problems and our main result is that the standard v cycle with point smoothing converges at a rate independent of the number of unknowns this is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric maxwell problems our multigrid analysis proceeds by applying the well known regularity based multigrid theory in order to apply this theory we prove regularity results for the axisymmetric laplace and maxwell equations in certain weighted sobolev spaces these together with some new finite element error estimates in certain weighted sobolev norms are the main ingredients of our analysis

exact solution of the schrodinger equation for a particle in a tetrahedral box the authors obtain the exact solution of the schrodinger equation for a particle confined to i an equilateral triangle ii a tetrahedral box with corners pi square root 2 pi square root 2 pi square root 2 pi square root 2 pi square root 2 pi 2 pi square root 2 pi square root 2 pi 2 and pi square root 2 pi square root 2 pi 2 the energies are for i enl 8h cross 2 pi 2 9ml2 n2 l2 ln where l is the side of the triangle and l n are distinct non zero integers and for ii elmn h cross 2 8m 3 l2 m2 n2 2lm 2mn 2nl where l m and n are distinct non zero integers the wavefunctions have been classified according to the irreducible representation of the corresponding symmetry groups

non sparse multiple kernel fisher discriminant analysis sparsity inducing multiple kernel fisher discriminant analysis mk fda has been studied in the literature building on recent advances in non sparse multiple kernel learning mkl we propose a non sparse version of mk fda which imposes a general lp norm regularisation on the kernel weights we formulate the associated optimisation problem as a semi infinite program sip and adapt an iterative wrapper algorithm to solve it we then discuss in light of latest advances in mkl optimisation techniques several reformulations and optimisation strategies that can potentially lead to significant improvements in the efficiency and scalability of mk fda we carry out extensive experiments on six datasets from various application areas and compare closely the performance of lp mk fda fixed norm mk fda and several variants of svm based mkl mk svm our results demonstrate that lp mk fda improves upon sparse mk fda in many practical situations the results also show that on image categorisation problems lp mk fda tends to outperform its svm counterpart finally we also discuss the connection between mk fda and mk svm under the unified framework of regularised kernel machines

a remark on kac wakimoto hierarchies of d type for the kac wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine lie algebra d 1 n we compute the coefficients of the corresponding hirota bilinear equations and verify the coincidence of these bilinear equations with the ones that are satisfied by givental s total descendant potential of the dn singularity as conjectured by givental and milanov 2005 simple singularities and integrable hierarchies the breadth of symplectic and poisson geometry prog math vol 232 boston birkh user pp 173 201

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

symmetric function products and plethysms and the boson fermion correspondence we use the boson fermion correspondence for s and q functions to establish some interesting properties concerning outer products and plethysms of s functions or q functions by power sum symmetric functions the techniques which are developed are also applied to computing the inverse kostka foulkes matrix which is the transition matrix between hall littlewood symmetric functions and s functions in some simple cases

rational solutions of integrable equations via nonlinear superposition formulae in this paper a method is developed to obtain rational solutions of integrable equations which include kdv boussinesq kp ito 1 1 dimensional cdgks ramani and 2 1 dimensional cdgks equations the key step that we use to find rational solutions of these equations is the use of nonlinear superposition formulae all these rational solutions obtained for each equation are connected by a backlund transformation which enables us to find other new solutions through the nonlinear superposition of rational solutions and other known solutions

schur functions and affine lie algebras abstract we make use of the representation theory of the infinite dimensional lie algebrasa b and formula to derive explicit formulas relating schur sp functions to schur ss functions

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

twisted vertex algebras bicharacter construction and boson fermion correspondences the boson fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics representation theory vertex algebras and conformal field theory integrable systems number theory cohomology two such correspondences are well known the types a and b and their super extensions as a main result of this paper we present a new boson fermion correspondence of type d a further we define a new concept of twisted vertex algebra of order n which generalizes super vertex algebra we develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras as well as for deriving formulas for the operator product expansions analytic continuations and normal ordered products by using the underlying hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples we show that the correspondences of types b c and d a are isomorphisms of twisted vertex algebras

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

a multifractal phase space analysis of perceptrons with biased patterns we calculate the multifractal spectrum of the partition of the coupling space of a perceptron induced by random input output pairs with non zero mean from the results we infer the influence of the input and output bias respectively on both the storage and generalization properties of the network it turns out that the value of the input bias is irrelevant as long as it is different from zero the generalization problem with output bias is new and shows an interesting two level scenario to compare our analytical results with simulations we introduce a simple and efficient algorithm to implement gibbs learning

polynomial tau functions of bkp and dkp hierarchies we construct all polynomial tau functions of the bkp dkp and mdkp hierarches we construct all polynomial tau functions of the bkp dkp and mdkp hierarches

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

gradient descent learning in perceptrons a review of its possibilities we present a streamlined formalism which reduces the calculation of the generalization error for a perceptron trained on random examples generated by a teacher perceptron to a matter of simple algebra the method is valid whenever the student perceptron can be identified as the unique minimum of a specific cost function the asymptotic generalization error is calculated explicitly for a broad class of cost functions and a specific cost function is singled out that leads to a generalization error extremely close to the one of the bayes classifier

new exact complex traveling wave solutions for 2 1 dimensional bkp equation by using some exact solutions of the auxiliary ordinary differential equation a direct algebraic method is obtained to construct the exact complex solutions for non linear partial differential equations the method is implemented for the 2 1 dimensional bkp equation new exact complex solutions are obtained

playing billiards in version space a ray tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples for randomly distributed examples the billiard estimate of the single perceptron with best average generalization probability agrees with known analytic results while for real life classification problems the generalization probability is consistently enhanced when compared to the maximal stability perceptron

bayes point machines kernel classifiers comprise a powerful class of non linear decision functions for binary classification the support vector machine is an example of a learning algorithm for kernel classifiers that singles out the consistent classifier with the largest margin i e minimal real valued output on the training sample within the set of consistent hypotheses the so called version space we suggest the bayes point machine as a well founded improvement which approximates the bayes optimal decision by the centre of mass of version space we present two algorithms to stochastically approximate the centre of mass of version space a billiard sampling algorithm and a sampling algorithm based on the well known perceptron algorithm it is shown how both algorithms can be extended to allow for soft boundaries in order to admit training errors experimentally we find that for the zero training error case bayes point machines consistently outperform support vector machines on both surrogate data and real world benchmark data sets in the soft boundary soft margin case the improvement over support vector machines is shown to be reduced finally we demonstrate that the real valued output of single bayes points on novel test points is a valid confidence measure and leads to a steady decrease in generalisation error when used as a rejection criterion

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy