moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

a new multicomponent ckp hierarchy and solutions based on the eigenfunction symmetry constraints we first propose a new multicomponent c type subhierarchy for kadomtsev petviashvili mcckp hierarchy and its lax representation which provides an effective way to find two types of new ckp equation with self consistent sources along with their related lax representations also it admits reductions to k constrained ckp hierarchy and to a 1 1 dimensional soliton hierarchy with self consistent sources which gives rise to two types of new kaup kuperschmidt kk equation with self consistent sources and two types of new bidirectional kaup kuperschmidt equation with self consistent sources by using the solutions of the ckp and kk equations and their corresponding eigenfunctions n soliton solutions for the first type of ckp self consistent source ckpscs and of kkscs are constructed by means of method of variation in constant respectively

nonparametric regression analysis achieved with topographic maps developed in combination with projection pursuit learning an application to density estimation and adaptive filtering of grey scale images a novel approach to nonparametric regression analysis using topographic maps is proposed the maps are trained with the extended maximum entropy learning rule emer in combination with projection pursuit regression ppr learning rather than a single map several maps are developed along optimally chosen projection directions in the input space in this way the regression performance improves in the case of sparsely sampled input spaces we explore two applications of the emer ppr combination 1 probability density estimation from pilot estimates and 2 adaptive filtering of grey scale images the first case is used as a testbed for comparing different both classic and neural network based regression techniques the results show that our emer ppr combination yields a superior regression performance for small data sets in the second case the regression model is trained on a noisy subimage the model obtained after training reduces the noise content of the full image by more than 20 db

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

periodic solutions of dynamical systems by a saddle point theorem of rabinowitz on montre comment par le theoreme de point selle rabinowitz on peut obtenir des resultats nouveaux et interessants sous des hypotheses simples

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

gradient based manipulation of nonparametric entropy estimates this paper derives a family of differential learning rules that optimize the shannon entropy at the output of an adaptive system via kernel density estimation in contrast to parametric formulations of entropy this nonparametric approach assumes no particular functional form of the output density we address problems associated with quantized data and finite sample size and implement efficient maximum likelihood techniques for optimizing the regularizer we also develop a normalized entropy estimate that is invariant with respect to affine transformations facilitating optimization of the shape rather than the scale of the output density kernel density estimates are smooth and differentiable this makes the derived entropy estimates amenable to manipulation by gradient descent the resulting weight updates are surprisingly simple and efficient learning rules that operate on pairs of input samples they can be tuned for data limited or memory limited situations or modified to give a fully online implementation

towards lax formulation of integrable hierarchies of topological type to each partition function of cohomological field theory one can associate an hamiltonian integrable hierarchy of topological type the givental group acts on such partition functions and consequently on the associated integrable hierarchies we consider the hirota and lax formulations of the deformation of the hierarchy of n copies of kdv obtained by an infinitesimal action of the givental group by first deforming the hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo differential operators we show that such deformed hierarchy admits an explicit lax formulation we then compare the deformed hamiltonians obtained from the lax equations with the analogous formulas obtained in buryak et al j differ geom 92 153 185 2012 buryak et al j geom phys 62 1639 1651 2012 to find that they agree we finally comment on the possibility of extending the hirota and lax formulation on the whole orbit of the givental group action

kernel based equiprobabilistic topographic map formation we introduce a new unsupervised competitive learning rule the kernel based maximum entropy learning rule kmer which performs equiprobabilistic topographic map formation in regular fixed topology lattices for use with nonparametric density estimation as well as nonparametric regression analysis the receptive fields of the formal neurons are overlapping radially symmetric kernels compatible with radial basis functions rbfs but unlike other learning schemes the radii of these kernels do not have to be chosen in an ad hoc manner the radii are adapted to the local input density together with the weight vectors that define the kernel centers so as to produce maps of which the neurons have an equal probability to be active equiprobabilistic maps both an online and a batch version of the learning rule are introduced which are applied to nonparametric density estimation and regression respectively the application envisaged is blind source separation bss from nonlinear noisy mixtures

gauge transformation and symmetries of the commutative multicomponent bkp hierarchy in this paper we defined a new multi component b type kadomtsev petviashvili bkp hierarchy that takes values in a commutative subalgebra of after this we give the gauge transformation of this commutative multicomponent bkp cmbkp hierarchy meanwhile we construct a new constrained cmbkp hierarchy that contains some new integrable systems including coupled kdv equations under a certain reduction after this the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi component bkp hierarchy will be constructed

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

lowner equations hirota equations and reductions of the universal whitham hierarchy this paper reconsiders finite variable reductions of the universal whitham hierarchy of genus zero in the perspective of dispersionless hirota equations in the case of one variable reduction dispersionless hirota equations turn out to be a powerful tool for understanding the mechanism of reduction all relevant equations describing the reduction lowner type equations and diagonal hydrodynamic equations can be thereby derived and justified in a unified manner the case of multi variable reductions is not so straightforward nevertheless the reduction procedure can be formulated in a general form and justified with the aid of dispersionless hirota equations as an application previous results of guil manas and martinez alonso are reconfirmed in this formulation

bispectral algebras of commuting ordinary differential operators we develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank n it combines and unifies the ideas of duistermaat grunbaum and wilson our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators the method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

characterizations of jacobians of curves with automorphisms we obtain a characterization of theta functions of jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties p p a v we first give a characterization in terms of finite dimensional orbits for a suitable action in the sato grassmannian secondly the introduction of formal baker akhiezer functions and formal t functions attached to a p p a v for the multipuncture case allows us to characterize in terms of bilinear identities those baker akhiezer functions that are baker akhiezer functions of jacobians of curves with automorphisms further in the case of automorphisms with fixed points we rewrite the previous result as a hierarchy of partial differential equations for the function of a p p a v finally since baker akhiezer and t functions are written in terms of theta functions these results give rise to characterizations of p p a v in terms of their theta functions

the morse index repeller attractor pairs and the connection index for semiflows on noncompact spaces abstract this paper extends the morse index theory of c c conley to semiflows on a noncompact meric space x is assumed to satisfy a hypothesis related to conditional contraction we collect background material define quasi index pairs and the morse index of a compact isolated invariant set k and prove that the morse index is a connected simple system we study repeller attractor pairs in k define index triples and prove their existence and several properties leading to the concepts of the connection index the connection map and the splitting class finally we consider paths continuous families of pairs k and study continuations of the morse and the connection indices along such paths the present paper is a sequel to the author s previous work on the homotopy index for infinite dimensional semiflows trans amer math soc 269 1982 351 382

periodic solutions of dynamical systems with bounded potential where ve c2 r x rn r v t t x y t x v t x jjbxiw y is bounded and v v t x 0 as 1x1 co for example we prove that if 0 d v r x m and v r x m as 1x1 co there exists at least one t periodic solution of v moreover if v j t 0 0 under suitable assumptions on v y t 0 this solution is not identically zero we also give existence results for the forced system

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

with central charge n we study representations of the central extension of the lie algebra of differential operators on the circle the algebra we obtain complete and specialized character formulas for a large class of representations which we call primitive these include all quasi finite irreducible unitary representations we show that any primitive representation with central chargen has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargen arise in this way we also establish a duality between integral modules of and finite dimensional irreducible modules ofgl n and conjecture their fusion rules

a lax representation for the vertex operator and the central extension integrable hierarchies viewed as isospectral deformations of an operatorl may admit symmetries they are time dependent vector fields transversal to and commuting with the hierarchy and forming an algebra in this work the commutation relations for the symmetries are shown to be based on a non commutative lie algebra splitting theorem the symmetries viewed as vector fields onl are expressed in terms of a lax pair

complete integrability of the kadomtsev petviashvili equation abstract the total hierarchy of the kadomtsev petviashvili kp equation is transformed to the system of linear partial differential equations with constant coefficients the complete integrability of the kp equation is proved by using this linear system the existence and uniqueness theorem of the cauchy problem of the kp hierarchy is obtained

on the lax representation of the 2 component kp and 2d toda hierarchies the lax formulation of the multicomponent kadomtsev petviashvili kp and 2d toda hierarchies involves several implicit constraints we show that at least in the 2 component case it is possible to explicitly solve such constraints and identify a set of free dependent variables for such hierarchies

the rpa equation embedded into infinite dimensional fock space f to clear up both algebraic and geometric structures for integrable systems derived from self consistent field theory in particular a geometrical aspect of the random phase approximation rpa equation is presented from the viewpoint of symmetry of the evolution equation the rpa equation for an infinite dimensional grassmannian is constructed it gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface

block algebra in two component bkp and d type drinfeld sokolov hierarchies we construct generalized additional symmetries of a two component bkp hierarchy defined by two pseudo differential lax operators these additional symmetry flows form a block type algebra with some modified or additional terms because of a b type reduction condition of this integrable hierarchy further we show that the d type drinfeld sokolov hierarchy which is a reduction of the two component bkp hierarchy possess a complete block type additional symmetry algebra that d type drinfeld sokolov hierarchy has a similar algebraic structure as the bigraded toda hierarchy which is a differential discrete integrable system

how to generate ordered maps by maximizing the mutual information between input and output signals a learning rule that performs gradient ascent in the average mutual information between input and an output signal is derived for a system having feedforward and lateral interactions several processes emerge as components of this learning rule hebb like modification and cooperation and competition among processing nodes topographic map formation is demonstrated using the learning rule an analytic expression relating the average mutual information to the response properties of nodes and their geometric arrangement is derived in certain cases this yields a relation between the local map magnification factor and the probability distribution in the input space the results provide new links between unsupervised learning and information theoretic optimization in a system whose properties are biologically motivated

darboux transformations for a 6 point scheme we introduce the jonas transformation a transformation of the darboux type for the general second order differential equation in two independent variables we present a discrete version of the transformation for a 6 point difference scheme the scheme is appropriate for the solution of a hyperbolic type initial boundary value problem we discuss several reductions and specifications of the transformations

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

new proof and generalizations of the demidowitsch schneider criterion a new proof of the demidowitsch schneider criterion on the absence of closed trajectories of r3 is given the new proof is generalized and applied in several directions rn vector fields n 3

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

non local matrix generalizations of w algebras there is a standard way to define two symplectic hamiltonian structures the first and second gelfand dikii brackets on the space of ordinarymth order linear differential opeatorsl d m u1dm 1 u2dm 2 u m in this paper i consider in detail the case where theu k aren n matrix valued functions with particular emphasis on the more interesting second gelfand dikii bracket of particular interest is the reduction to the symplectic submanifoldu1 0 this reduction gives rise to matrix generalizations of the classical version of thenon linear w m algebras calledv n m algebras the non commutativity of the matrices leads tonon local terms in thesev n m algebra s i show that these algebras contain a conformal virasoro subalgebra and that combinationsw k of theu k can be formed that aren n matrices of conformally primary fields of spink in analogy with the scalar casen 1 in general however thev m n algebras have a much richer structure than thew m algebras as can be seen on the examples of thenon linear andnon local poisson brackets u2 ab u2 cd u2 ab w3 cd and w3 ab w3 cd which i work out explicitly for allm andn a matrix miura transformations is derived mapping these complicated second gelfand dikii brackets of theu k to a set of much simpler poisson brackets providing the analogoue of the free field representation of thew m algebras

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

pfaff tau functions consider the two dimensional toda lattice with certain skew symmetric initial condition which is preserved along the locus s t of the space of time variables restricting the solution to s t we obtain another hierarchy called pfaff lattice which has its own tau function being equal to the square root of the restriction of 2d toda tau function we study its bilinear and fay identities w and virasoro symmetries relation to symmetric and symplectic matrix integrals and quasiperiodic solutions

the w1 gls symmetries of the s component kp hierarchy adler shiota and van moerbeke obtained for the kp and toda lattice hierarchies a formula which translates the action of the vertex operator on tau functions to an action of a vertex operator of pseudodifferential operators on wave functions this relates the additional symmetries of the kp and toda lattice hierarchy to the w1 respectively w1 w1 algebra symmetries in this paper we generalize the results to the s component kp hierarchy the vertex operators generate the algebra w1 gls the matrix version of w1 since the toda lattice hierarchy is formally equivalent to the 2 component kp hierarchy the results of this article uncover in that particular case a much richer structure than the one obtained by adler shiota and van moerbeke

solvability of the super kp equation and a generalization of the birkhoff decomposition the unique solvability of the initial value problem for the total hierarchy of the super kadomtsev petviashvili system is established to prove the existence we use a generalization of the birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro i e pseudo differential operators to show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space the usual kp system with non commutative coefficients is also studied the kp system is obtained from the super kp system by reduction modulo odd variables on the other hand the first modified kp equation can be obtained from the super kp system by elimination of odd variables thus the super kp system is a natural unification of the kp system and the modified kp systems

the formation of topographic maps that maximize the average mutual information of the output responses to noiseless input signals this article introduces an extremely simple and local learning rule for to pographic map formation the rule called the maximum entropy learning rule mer maximizes the unconditional entropy of the map s output for any type of input distribution the aim of this article is to show that mer is a viable strategy for building topographic maps that maximize the average mutual information of the output responses to noiseless input signals when only input noise and noise added input signals are available

a generalized dressing approach for solving the extended kp and the extended mkp hierarchy a combination of dressing method and variation of constants as well as a formula for constructing the eigenfunction is used to solve the extended kp hierarchy which is a hierarchy with one more series of time flow and based on the symmetry constraint of kp hierarchy similarly extended mkp hierarchy is formulated and its zero curvature form lax representation and reductions are presented via gauge transformation it is easy to transform dressing solutions of extended kp hierarchy to the solutions of extended mkp hierarchy wronskian solutions of extended kp and extended mkp hierarchies are constructed explicitly

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

givental symmetries of frobenius manifolds and multi component kp tau functions we establish a link between two different constructions of the action of the twisted loop group on the space of frobenius structures the first construction due to givental describes the action of the twisted loop group on the partition functions of formal axiomatic gromov witten theories the explicit formulas for the corresponding tangent action were computed by y p lee the second construction due to van de leur describes the action of the same group on the space of frobenius structures via the multi component kp hierarchies our main theorem states that the genus zero restriction of the y p lee formulas coincides with the tangent van de leur action

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

solutions of the wdvv equations and integrable hierarchies of kp type we show that reductions of kp hierarchies related to the loop algebra of sl n with homogeneous gradation give solutions of the darboux egoroff system of pde s using explicit dressing matrices of the riemann hilbert problem generalized to include a set of commuting additional symmetries we construct solutions of the witten dijkgraaf e verlinde h verlinde equations

universal whitham hierarchy dispersionless hirota equations and multicomponent kp hierarchy abstract the goal of this paper is to identify the universal whitham hierarchy of genus zero with a dispersionless limit of the multicomponent kp hierarchy to this end the multicomponent kp hierarchy is re formulated to depend on several discrete variables called charges these discrete variables play the role of lattice coordinates in underlying toda field equations a multicomponent version of the so called differential fay identity are derived from the hirota equations of the function of this charged multicomponent kp hierarchy these multicomponent differential fay identities have a well defined dispersionless limit the dispersionless hirota equations the dispersionless hirota equations turn out to be equivalent to the hamilton jacobi equations for the s functions of the universal whitham hierarchy the differential fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar valued wave functions of the multicomponent kp hierarchy

multiple solutions of the periodic boundary value problem for some forced pendulum type equations if f x c 0 and e is a constant t then integrating 0 1 over 0 272 after multiplication of both members of the equation by x shows that the only possible solutions are constant and hence satisfy a sin x c consequently if we exclude the trivial translation by a multiple of 275 0 1 will have no solution if dj a one solution if lp a and two solutions if lc l a in 0th er t erms the intersection of the range of the operator d2 dt2 c d dt f u sin acting on zn periodic functions of class g with the subspace of constant functions in the space c o 2711 of real continuous functions on 0 27r is the closed interval a a whose interior points are images of two distinct solutions and boundary points of one we try to obtain similar results for the case where e is no more constant and

the q deformed mkp hierarchy with self consistent sources wronskian solutions and solitons based on the eigenfunction symmetry constraint of the q deformed modified kp hierarchy a q deformed mkp hierarchy with self consistent sources q mkphscss is constructed the q mkphscss contain two types of q deformed mkp equation with self consistent sources by the combination of the dressing method and the method of variation of constants a generalized dressing approach is proposed to solve the q deformed kp hierarchy with self consistent sources q kphscss using the gauge transformation between the q kphscss and the q mkphscss the q deformed wronskian solutions for the q kphscss and the q mkphscss are obtained the one soliton solutions for the q deformed kp mkp equation with a source are given explicitly

dispersionful analog of the whitham hierarchy the transition from the zero genus universal whitham hierarchy to its dispersionful counterpart making use only of the lax representations is presented this is an alternative approach to that of takasaki who has recently shown that the dispersionless limit of the charged multicomponent kadomtsev petviashvili kp hierarchy is the whitham hierarchy the advantage of the presented approach is the construction of finite field reductions which are the main concern in this paper the theory is illustrated by several significant examples

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

schlesinger backlund transformations for the n component kp the n component kadomtsev petviashvili kp hierarchy was introduced by sato as the set of deformation equations i e lax equations of n 1n n matrix pseudodifferential operators on such a set of operators we define a schlesinger backlund transformation which produces from a set of solutions of the kp hierarchy a new set of solutions applying such transformations recursively we obtain a whole family of solutions from which we can then derive the kp tau function

backlund darboux transformations for the coupled kp hierarchy using the geometry of the infinite isotropic grassmannian related to the hirota ohta coupled kp hierarchy we construct its elementary backlund darboux transformations

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

topological strings and integrable hierarchies we consider the topological b model on local calabi yau geometries we show how one can solve for the amplitudes by using w algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the calabi yau in the highly effective fermionic brane formulation this leads to a free fermion description of the amplitudes furthermore we argue that topological strings on calabi yau geometries provide a unifying picture connecting non critical super strings integrable hierarchies and various matrix models in particular we show how the ordinary matrix model the double scaling limit of matrix models and kontsevich like matrix model are all related and arise from studying branes in specific local calabi yau three folds we also show how a model topological string on p 1 and local toric threefolds and in particular the topological vertex can be realized and solved as b model topological string amplitudes on a calabi yau manifold

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

modified non abelian toda field equations and twisted quasigraded lie algebras we construct a new family of quasigraded lie algebras that admit the kostant adler scheme they coincide with special quasigraded deformations of twisted subalgebras of the loop algebras using them we obtain new hierarchies of integrable equations in partial derivatives which we call modified non abelian toda field hierarchies

a new extended discrete kp hierarchy and a generalized dressing method inspired by the squared eigenfunction symmetry constraint we introduce a new k flow by extending a specific tn flow of a discrete kp hierarchy dkph we construct an extended discrete kph exdkph which consists of tn flow k flow and tn evolution of eigenfunction and adjoint eigenfunctions and its lax representation the exdkph contains two types of discrete kp equation with self consistent sources dkpescs two reductions of exdkph are obtained the generalized dressing approach for solving the exdkph is proposed and the n soliton solutions of two types of the dkpescs are presented

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

cooperative information maximization with gaussian activation functions for self organizing maps in this paper we propose a new information theoretic method to produce explicit self organizing maps soms competition is realized by maximizing mutual information between input patterns and competitive units competitive unit outputs are computed by the gaussian function of distance between input patterns and competitive units a property of this gaussian function is that as distance becomes smaller a neuron tends to fire strongly cooperation processes are realized by taking into account the firing rates of neighboring neurons we applied our method to uniform distribution learning chemical compound classification and road classification experimental results confirmed that cooperation processes could significantly increase information content in input patterns when cooperative operations are not effective in increasing information mutual information as well as entropy maximization is used to increase information experimental results showed that entropy maximization could be used to increase information and to obtain clearer soms because competitive units are forced to be equally used on average