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

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

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

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

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

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

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

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

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

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

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

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

comments on a weighted least squares method for the design of stable 1 d and 2 d iir digital filters with reply for original paper by lu et al see ieee trans signal processing vol 46 p 1 10 1998 january the iterative procedure by lu et al for designing iir filters is recognized as the sanathanan koerner or steiglitz mcbride iteration which was proposed originally in 1963 and 1965 respectively for system identification purposes we re examine some claims and issues related to lu et al s paper in view of various properties that have been deduced for the steiglitz mcbride iteration in the identification literature in particular the steiglitz mcbride procedure is known not to minimize a least squares distance measure contrary to lu et al s claim it can however provide good models and remain stable during successive iterations under specific conditions which are reviewed a reply is given by lu et al

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

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

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

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

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 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