multivariate hypergeometric functions as τ functions of toda lattice and kadomtsev petviashvili equation we present the q deformed multivariate hypergeometric functions related to schur polynomials as tau functions of the kp and of the two dimensional toda lattice hierarchies the variables of the hypergeometric functions are the higher times of those hierarchies the discrete toda lattice variable shifts parameters of hypergeometric functions the role of additional symmetries in generating hypergeometric tau functions is explained

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

milne s hypergeometric functions in terms of free fermions we present the fermionic representation for the q deformed hypergeometric functions related to schur polynomials we show that these multivariate hypergeometric functions are tau functions of the kp hierarchy and at the same time they are the ratios of toda lattice tau functions considered by takasaki evaluated at certain values of higher toda lattice times the variables of the hypergeometric functions are related to the higher times of those hierarchies via a miwa change of variables the discrete toda lattice variable shifts parameters of hypergeometric functions hypergeometric functions of type p s can also be viewed as a group 2 cocycle for the do on the circle the group times are higher times of tl hierarchy and the arguments of a hypergeometric function we obtain the determinant representation and the integral representation of a special type of kp tau functions these results generalize some of the results of milne concerning multivariate hypergeometric functions we write down a system of partial differential equations for these tau functions string equations

dressing method darboux transformation and generalized restricted flows for the kdv hierarchy abstract we define generalized restricted flows by restricting general symmetry of the kdv hierarchy lax representation for generalized restricted flows is derived with the use of infinitesimal version of the zakharov shabat method finite form of this method is employed for deriving darboux transformation for restricted flows this darboux transformation adds one degree of freedom

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

on the infinite dimensional symmetry group of the davey stewartson equations the lie algebra of the group of point transformations leaving the davey stewartson equations dse s invariant is obtained the general element of this algebra depends on four arbitrary functions of time the algebra is shown to have a loop structure a property shared by the symmetry algebras of all known 2 1 dimensional integrable nonlinear equations subalgebras of the symmetry algebra are classified and used to reduce the dse s to various equations involving only two independent variables

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

large deviations for non uniformly expanding maps we obtain large deviation bounds for non uniformly expanding maps with non flat singularities or criticalities and for partially hyperbolic non uniformly expanding attracting sets that is given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map showing that the lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved as easy by products we deduce escape rates from subsets of the basins of physical measures for these types of maps the rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states

on the structure of unitary conformal field theory ii representation theoretic approach two dimensional unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras chiral algebras are equipped with a family of co multiplications which serve to define tensor product representations chiral vertices arise as clebsch gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra the algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by yang baxter braid matrices chiral fusion is analyzed it is shown that the braid and fusion matrices determine invariants of knots and links connections between the representation theories of chiral algebras and of quantum groups are sketched finally it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras

duality and modular invariance in rational conformal field theories we investigate the polynomial equations which should be satisfied by the duality data for a rational conformal field theory we show that by these duality data we can construct some vector spaces which are isomorphic to the spaces of conformal blocks one can construct explicitly the inner product for the former if one deals with a unitary theory these vector spaces endowed with an inner product are the algebraic reminiscences of the hilbert spaces in a chern simons theory as by products we show that the polynomial equations involving the modular transformations for the one point blocks on the torus are not independent we discuss the solution of structure constants for a physical theory making some assumption we obtain a neat solution and this solution in turn implies that the quantum groups of the left sector and of the right sector must be the same although the chiral algebras need not be the same some examples are given finally we discuss the reconstruction of the quantum group in a rational conformal theory

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

pesin s entropy formula for c 1 c 1 non uniformly expanding maps we prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials moreover we formulate a c 1 generalization of pesin s entropy formula all ergodic weak srb like measures satisfy pesin s entropy formula for c 1 non uniformly expanding maps we show that for weak expanding maps such that text leb a e x has positive frequency of hyperbolic times then all the necessarily existing ergodic weak srb like measures satisfy pesin s entropy formula and are equilibrium states for the potential psi log det df in particular this holds for any c 1 expanding map and in this case the set of invariant probability measures that satisfy pesin s entropy formula is the weak closed convex hull of the ergodic weak srb like measures

the additional symmetries for the btl and ctl hierarchies in this paper we construct the additional symmetries for the toda lattice hierarchies of b type and c type the btl and ctl hierarchies and show their algebraic structures are w b w b and w c w c respectively and also we discuss the generating functions of the additional symmetries

new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

dispersionless toda hierarchy and two dimensional string theory the dispersionless toda hierarchy turns out to lie in the heart of a recently proposed landau ginzburg formulation of two dimensional string theory at self dual compactification radius the dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy this solution is obtained by solving a riemann hilbert problem equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless toda hierarchy fundamental ingredients of the landau ginzburg formulation such as landau ginzburg potentials and tachyon landau ginzburg fields are translated into the language of the lax formalism furthermore a wedge algebra is pointed out to exist behind the riemann hilbert problem and speculations on its possible role as generators of extra states and fields are presented

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

generalized ablowitz ladik hierarchy in topological string theory this paper addresses the issue of integrable structures in topological string theory on generalized conifolds open string amplitudes of this theory can be expressed as the matrix elements of an operator on the fock space of 2d charged free fermion fields the generating function of these amplitudes with respect to the product of two independent schur functions becomes a tau function of the 2d toda hierarchy the associated lax operators turn out to have a particular factorized form this factorized form of the lax operators characterizes a generalization of the ablowitz ladik hierarchy embedded in the 2d toda hierarchy the generalized ablowitz ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds

existence and uniqueness of srb measure on c 1 generic hyperbolic attractors let m be a smooth riemannian manifold we show that for c1 generic f in rm diff 1 m if f has a hyperbolic attractor f then there exists a unique srb measure supported on f moreover the srb measure happens to be the unique equilibrium state of potential function psi f in c 0 lambda f defined by psi f x log det df e u x x in lambda f where e u x is the unstable space of txm

locality of conformal fields in two dimensions exchange algebra on the light cone we discuss the exchange algebra of light cone operators as the fundamental structure of two dimensional conformal quantum field theory it is necessary in order to account for the locality properties of wightman functions of conformal fields we discuss the consistency requirements of this new type of algebra and obtain a classification containing the well known minimal models

boundaries of markov partitions the core of a markov partition is the nonwandering set of the map restricted to the boundary of the partition we show that the core of a markov partition is always a finitely presented system then we show that every one sided sofic system occurs as the core of a markov partition for an n fold covering map on the circle and every two sided sofic system occurs as the core of a markov partition for a hyperbolic automorphism of the two dimensional torus

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

perturbation theory around two dimensional critical systems through holomorphic decomposition the authors consider perturbations by relevant interactions of two dimensional conformally invariant rational field theories the meromorphic structure with respect to the scaling dimensions of the perturbing interactions of the correlation functions is made explicit through a successive application of stoke s theorem the resulting decomposition of the amplitudes into holomorphic and antiholomorphic factors yields a representation of the meromorphic structure in terms of the basic data of the rational field theory which are scaling dimensions operator product coefficients and braid matrices they exemplify the general deduction by a concrete calculation to the third order in the coupling constant of the perturbing interaction

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

quadratic maps without asymptotic measure an interval map is said to have an asymptotic measure if the time averages of the iterates of lebesgue measure converge weakly we construct quadratic maps which have no asymptotic measure we also find examples of quadratic maps which have an asymptotic measure with very unexpected properties e g a map with the point mass on an unstable fix point as asymptotic measure the key to our construction is a new characterization of kneading sequences

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

infinite dimensional grassmannian structure of two dimensional quantum gravity we study the infinite dimensional grassmannian structure of 2d quantum gravity coupled to minimal conformal matters and show that there exists a large symmetry thew 1 symmetry using this symmetry structure we prove that the square root of the partition function which is a function of thep reduced kp hierarchy satisfies the vacuum condition of thew 1 algebra we further show that this condition is reduced to the vacuum condition of thew p algebra when the redundant variables for thep reduction are eliminated this mechanism also gives a prescription for extracting thew p algebra from thew 1 algebra

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

integrable nonlinear evolution equations with time dependent coefficients a simple and straightforward method for generating completely integrable nonlinear evolution equations with time dependent coefficients is presented herein for the equations under consideration the solutions to those given by vector fields which are independent of time are given thus explicit links between equations are obtained as an application of the proposed method it is shown that the linear superposition with arbitrary time dependent coefficients of different members of an integrable hierarchy is again integrable furthermore it turns out that for some integrable equations like the korteweg de vries kdv the benjamin ono bo or the kadomtsev petviashvili kp the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow those flows demonstrated for the kdv which can be generated by lie homomorphisms coming from first order problems are completely classified many well known equations which can be found in the literature are of that type as an application of such a first order link a direct link is given from kdv to the cylindrical kdv and from there to the kp with nontrivial dependence on the second spatial variable

toda lattice hierarchy and generalized string equations string equations of thepth generalized kontsevich model and the compactifiedc 1 string theory are re examined in the language of the toda lattice hierarchy as opposed to a hypothesis postulated in the literature the generalized kontsevich model atp 1 does not coincide with thec 1 string theory at selfdual radius a broader family of solutions of the toda lattice hierarchy including these models is constructed and shown to satisfy generalized string equations the status of a variety ofc 1 string models is discussed in this new framework

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

on additional symmetries of the kp hierarchy and sato s backlund transformation a short proof is given to the fact that the additional symmetries of the kp hierarchy defined by their action on pseudodifferential operators according to fuchssteiner chen lee lin orlov shulman coincide with those defined by their action on functions as sato s backlund transformations a new simple formula for the generator of additional symmetries is also presented

a matrix integral solution to two dimensional w p gravity thep th gel fand dickey equation and the string equation l p 1 have a common solution expressible in terms of an integral overn n hermitean matrices for largen the integrand being a perturbation of a gaussian generalizing kontsevich s integral beyond the kdv case it is equivalent to showing that is a vacuum vector for aw generated from the coefficients of the vertex operator this connection is established via a quadratic identity involving the wave function and the vertex operator which is a disguised differential version of the fay identity the latter is also the key to the spectral theory for the two compatible symplectic structures of kdv in terms of the stress energy tensor associated with the virasoro algebra given a differential operator l d p q 2 t d p 2 cdots q p t with d frac partial dx t t 1 t 2 t 3 x equiv t 1 consider the deformation equations1 0 1 begin gathered frac partial l partial t n l n p l n 1 2 n 0 mod p hfill p reduced kp equation hfill end gathered ofl for which there exists a differential operatorp possibly of infinite order such that 0 2 l p 1 string equation in this note we give a complete solution to this problem in section 1 we give a brief survey of useful facts about thei function t the wave function t z solution of t n l n p x andl 1 p z and the corresponding infinitedimensional planev 0 of formal power series inz for largez v 0 span psi t z for all t in mathbb c infty in sato s grassmannian the three theorems below form the core of the paper their proof will be given in subseuqent sections each of which lives on its own right

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

equations of arbitrary order invariant under the kadomtsev petviashvili symmetry group by means of a simple new approach a general kadomtsev petviashvili kp family with an arbitrary function of group invariants of arbitrary order is proposed it is proved that the general kp family possesses a common infinite dimensional kac moody virasoro lie point symmetry algebra the known fourth order one can be re obtained as a special example the finite transformation group is presented in a clearer form the kac moody virasoro group invariant solutions and the kac moody group invariant solutions of the kp family are determined by the boussinesq and kdv families respectively

conformal and lie superalgebras motivated from free fermionic fields in this paper we construct six families of conformal superalgebras of infinite type motivated from free quadratic fermonic fields with derivatives and we prove their simplicity the lie superalgebras generated by these conformal superalgebras are proven to be simple except for a few special cases in the general linear superalgebras and the type q lie superalgebras in which these lie superalgebras have a one dimensional centre and the quotient lie superalgebras modulo the centre are simple certain natural central extensions of these families of conformal superalgebras are also given moreover we prove that these conformal superalgebras are generated by their finite dimensional subspaces of minimal weight in a certain sense it is shown that a conformal superalgebra is simple if and only if its generated lie superalgebra does not contain a proper nontrivial ideal with a one variable structure

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

on conserved densities and asymptotic behaviour for the potential kadomtsev petviashvili equation we study local conservation laws with non vanishing conserved densities and corresponding boundary conditions for the potential kadomtsev petviashvili equation we analyse an infinite symmetry group of the equation and generate a finite number of conserved densities corresponding to infinite symmetries through appropriate boundary conditions

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

the exchange algebra for zamolodchikov and fateev s parafermionic theories the concept of an exchange algebra has recently been introduced by rehren and schroer 1989 in the context of two dimensional conformal field theories to give an algebraic setting to both the dynamics and the locality requirement labelling the conformal families with two indices and assuming an interpolating scheme for one of the fields it is shown that the braiding matrices for a subset of fields in zamolodchikov s and fateev s 1986 parafermionic theories containing all the order parameters are identical to those of the diagonal minimal models the authors recover the full spectrum of these theories modulo integers from the phase condition of the exchange algebra even though the subset does not include the parafermionic currents

are all the equations of the kadomtsev petviashvili hierarchy integrable the kadomtsev petviashvili kp hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the kp hierarchy provides integrable scalar equations with an arbitrary number of independent variables it is shown by investigating a specific equation in 3 1 dimensions that the higher equations in the kp hierarchy are only integrable in a conditional sense the equation under study taken in isolation does not pass certain well known and reliable integrability tests thus applying painleve analysis we find that solutions exist allowing movable critical points furthermore solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions on the other hand if the dependence of a solution on the first 2 1 variables is restricted by the fact that it should also satisfy the kp equation itself then the integrability conditions in the other dimensions are satisfied conditional integrability thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy

some constraints and solutions of the kadomtsev petviashvili equation we illustrate the relations among the symmetry invariant group and constraint for a differential equation applied to the kadomtsev petviashvili equation some constraints and solutions are given in particular the equation associated with symmetry sigma of the kp equation is introduced and discussed

on pesin s entropy formula for dominated splittings without mixed behavior abstract for c 1 diffeomorphisms we prove that the pesin s entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior we also prove shub s entropy conjecture for diffeomorphisms having such kind of splittings

toda hierarchies and their applications the 2d toda hierarchy occupies a central position in the family of integrable hierarchies of the toda type the 1d toda hierarchy and the ablowitz ladik aka relativistic toda hierarchy can be derived from the 2d toda hierarchy as reductions these integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s a recent example is a series of studies on models of statistical mechanics called the melting crystal model this research has revealed that the aforementioned two reductions of the 2d toda hierarchy underlie two different melting crystal models technical clues are a fermionic realization of the quantum torus algebra special algebraic relations therein called shift symmetries and a matrix factorization problem the two melting crystal models thus exhibit remarkable similarity with the hermitian and unitary matrix models for which the two reductions of the 2d toda hierarchy play the role of fundamental integrable structures

srb measures and pesin s entropy formula for endomorphisms a celebrated result conjectured by ruelle and proved by ledrappier strelcyn and young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a c 2 diffeomorphism satisfies pesin s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds 41 42 the later property is known as srb property of the invariant measure this result was also successfully generalized to random dynamical systems of diffeomorphisms 51

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

generalized kadomtsev petviashvili equation with an infinite dimensional symmetry algebra abstract a generalized kadomtsev petviashvili equation describing water waves in oceans of varying depth density and vorticity is discussed a priori it involves 9 arbitrary functions of one or two variables the conditions are determined under which the equation allows an infinite dimensional symmetry algebra this algebra can involve up to three arbitrary functions of time it depends on precisely three such functions if and only if it is completely integrable

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

algebra of pseudodifferential operators and symmetries of equations in the kadomtsev petviashvili hierarchy point symmetries are obtained for all equations in the kp hierarchy the lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tn the symmetry algebra is a semidirect sum of a virasoro algebra and a kac moody one the positive part of this algebra is embedded into the known w algebra of kp symmetries and into the free fermion algebra l the corresponding action on the tau function is presented the negative part of the point symmetries does not fit into the free fermion algebra but is embedded into a p algebra based on the algebra of pseudodifferential operators

the ghost symmetry of the bkp hierarchy in this paper we systematically develop the ghost symmetry of the kadomtsev petviashvili sub hierarchy of b type bkp through its actions on the lax operator l the eigenfunctions and the function in this process the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the bkp hierarchy moreover the bilinear identity of the constrained bkp hierarchy and adler shiota van moerbeke formula of the bkp hierarchy are rederived compactly by means of the spectral representation and ghost symmetry

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

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

differential equations in the spectral parameter darboux transformations and a hierarchy of master symmetries for kdv we study a certain family of schrodinger operators whose eigenfunctions satisfy a differential equation in the spectral parameter of the formb x we show that the flows of a hierarchy of master symmetries for kdv are tangent to the manifolds that compose the strata of this class ofbispectral potentials this extends and complements a result of duistermaat and grunbaum concerning a similar property for the adler and moser potentials and the flows of the kdv hierarchy

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property

on the constrained b type kadomtsev petviashvili hierarchy hirota bilinear equations and virasoro symmetry we derive the bilinear equations of the constrained bkp hierarchy from the calculus of pseudodifferential operators the full hierarchy equations can be expressed in hirota s bilinear form characterized by the functions and besides we also give a modification of the original orlov schulman additional symmetry to preserve the constrained form of the lax operator for this hierarchy the vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless virasoro algebra

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

method of squared eigenfunction potentials in integrable hierarchies of kp type the method of squared eigenfunction potentials sep is developed systematically to describe and gain new information about the kadomtsev petviashvili kp hierarchy and its reductions interrelation to the function method is discussed in detail the principal result which forms the basis of our sep method is the proof that any eigenfunction of the general kp hierarchy can be represented as a spectral integral over the baker akhiezer ba wave function with a spectral density expressed in terms of sep in fact the spectral representations of the adjoint ba functions can in turn be considered as defining equations for the kp hierarchy the sep method is subsequently used to show how the reduction of the full kp hierarchy to the constrained kp ckp rm hierarchies can be given entirely in terms of linear constraint equations on the pertinent functions the concept of sep turns out to be crucial in providing a description of ckp rm hierarchies in the language of the universal sato grassmannian and finding the non isospectral virasoro symmetry generators acting on the underlying functions the sep method is used to write down generalized binary darboux backlund transformations for constrained kp hierarchies whose orbits are shown to correspond to a new toda model on a square lattice as a result we obtain a series of new determinant solutions for the functions generalizing the known wronskian multi soliton solutions finally applications to random matrix models in condensed matter physics are briefly discussed

on the kms condition for chiral conformal field theories using the kubo martin schwinger kms condition and exchange algebras we discuss how to compute correlation functions for the conformal exchange field theories in particular we derive the verlinde algebra from modular invariance of the conformal field theories we also show how the verlinde matrices can be related to the kms matrices

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

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

thermodynamic limit of random partitions and dispersionless toda hierarchy we study the thermodynamic limit of random partition models for the instanton sum of 4d and 5d supersymmetric u 1 gauge theories deformed by some physical observables the physical observables correspond to external potentials in the statistical model the partition function is reformulated in terms of the density function of maya diagrams the thermodynamic limit is governed by a limit shape of young diagrams associated with dominant terms in the partition function the limit shape is characterized by a variational problem which is further converted to a scalar valued riemann hilbert problem this riemann hilbert problem is solved with the aid of a complex curve which may be thought of as the seiberg witten curve of the deformed u 1 gauge theory this solution of the riemann hilbert problem is identified with a special solution of the dispersionless toda hierarchy that satisfies a pair of generalized string equations the generalized string equations for the 5d gauge theory are shown to be related to hidden symmetries of the statistical model the prepotential and the seiberg witten differential are also considered

generalized bounded variation and applications to piecewise monotonic transformations we prove the quasi compactness of the perron frobenius operator of piecewise monotonic transformations when the inverse of the derivative is holder continuous or more generally of bounded p variation

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

negative flows of the potential kp hierarchy we construct a grassmannian like formulation for the potential kp hierarchy including additional negative flows our approach will generalize the notion of a r function to include negative flows we compare the resulting hierarchy with results by hirota satsuma and bogoyavlenskii

point symmetries of conditionally integrable nonlinear evolution equations the lie point symmetries of the first two equations in the kadomtsev petviashvili kp hierarchy introduced by jimbo and miwa are investigated the first is the potential kp equation the second involves four independent variables and is called the jimbo miwa jm equation the joint symmetry algebra for the two equations is shown to have a kac moody virasoro structure whereas the symmetry algebra of the jm equation alone does not subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions

symmetries and reductions on the noncommutative kadomtsev petviashvili and gelfand dickey hierarchies in this paper we construct the additional flows of the noncommutative kadomtsev petviashvili kp hierarchy and the additional symmetry flows constitute an infinite dimensional lie algebra w1 in addition the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative kp hierarchy with self consistent sources and the constrained noncommutative kp hierarchy the above results will be further generalized to the noncommutative gelfand dickey hierarchies which contain many interesting noncommutative integrable systems such as the noncommutative kdv hierarchy and noncommutative boussinesq hierarchy meanwhile we construct two new noncommutative systems including odd noncommutative c type gelfand dickey and even noncommutative c type gelfand dickey hierarchies also using the symmetry we can construct a new noncommutative gelfand dickey hierarchy with self consistent sources based on the natural differential lax operator of the noncommutative gelfand dickey hierarchy the string equations of the noncommutative gelfand dickey hierarchy are also derived in this paper we construct the additional flows of the noncommutative kadomtsev petviashvili kp hierarchy and the additional symmetry flows constitute an infinite dimensional lie algebra w1 in addition the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative kp hierarchy with self consistent sources and the constrained noncommutative kp hierarchy the above results will be further generalized to the noncommutative gelfand dickey hierarchies which contain many interesting noncommutative integrable systems such as the noncommutative kdv hierarchy and noncommutative boussinesq hierarchy meanwhile we construct two new noncommutative systems including odd noncommutative c type gelfand dickey and even noncommutative c type gelfand dickey hierarchies also using the symmetry we can construct a new noncommutative gelfand dickey hierarchy with self consistent sources based o

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

braiding matrices modular transformations and topological field theories in 2 1 dimensions relations between 3d topological field theories and rational conformal field theories are discussed in the former framework we can define the generalized verlinde operators using these operators we find modular transformations for conformal blocks of one point functions and two point functions on the torus the result is generalized to higher genus the correctness of our formulae is illustrated by some examples we also emphasize the importance of the fusion algebra

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

symmetry reductions of the lax pair for the 2 1 dimensional konopelchenko dubrovsky equation based on the symbolic computational system maple the similarity reduction arising from the classical lie point symmetries of the lax pair for the 2 1 dimensional konopelchenko dubrovsky kd equation is carried out we obtain several interesting reductions by analyzing not only the reduced lax pair but also the kd equation reduced under the same symmetry group we find that the reduced lax pairs do not always lead to the reduced kd equation

toward a canonical formalism of non perturbative two dimensional gravity on the basis of the previously proposed action principle describing the theory space of 2d gravity in less than one dimension we develop a systematic canonical formalism for studying the properties of the string equation in the phase space of the cosmological constant and its canonical conjugate the puncture operator the string equation is written in a manifestly invariant form under the group of regular canonical transformations in the phase space of generalized coordinate and momentum as a consequence the geometrical origin of the generalized virasoro condition on the partition function or more precisely the function is understood to be the symmetry under the regular area preserving diffeomorphisms w1 symmetry in the deformed phase space the deformed canonical formalism can be regarded as a quantization of a classical canonical formalism describing the sphere limit of the theory

the quantum group structure of 2d gravity and minimal models ii the genus zero chiral bootstrap the chiral operator algebra of the quantum group covariant operators of vertex type is completely worked out by making use of the operator approach suggested by the liouville theory where the quantum group symmetry is explicit this completes earlier articles along the same line the relationship between the quantum group invariant of irf type and quantum group covariant of vertex type chiral operator algebras is fully clarified and connected with the transition to the shadow world for quantum group symbols the corresponding 3 j symbol dressing is shown to reduce to the simpler transformation of babelon and one of the authors j l g in a suitable infinite limit defined by analytic continuation the above two types of operators are found to coincide when applied to states with liouville momenta going to in a suitable way the introduction of quantum group covariant operators in the three dimensional picture gives a generalization of the quantum group version of discrete three dimensional gravity that includes tetrahedra associated with 3 j symbols and universalr matrix elements altogether the present work and a previous parallel article gives the concrete realization of moore and seiberg s scheme that describes the chiral operator algebra of two dimensional gravity and minimal models

analysis of the symmetry group and exact solutions of the dispersionless kp equation in n 1 dimensions the lie algebra of the symmetry group of the n 1 dimensional generalization of the dispersionless kadomtsev petviashvili equation is obtained and identified as a semi direct sum of a finite dimensional simple lie algebra and an infinite dimensional nilpotent subalgebra group transformation properties of solutions under the subalgebra sl 2 r are presented known explicit analytic solutions in the literature are shown to be actually group invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations

on a new hierarchy of symmetries for the kadomtsev petviashvili equation abstract we present a new hierarchy of symmetries for the kadomtsev petviashvili equation these new symmetries depend on the space and time variables explicitly together with the previously known classical symmetries they constitute an infinite dimensional lie algebra