generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

alternative descriptions and bipartite compound quantum systems we analyse some features of alternative hermitian and quasi hermitian quantum descriptions of simple and bipartite compound systems we show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as a tensor product of positive operators on component spaces some examples also show that such property could be strictly connected with symmetry properties of the non hermitian hamiltonian

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

extensions of the matrix gelfand dickey hierarchy from generalized drinfeld sokolov reduction thep p matrix version of ther kdv hierarchy has been recently treated as the reduced system arising in a drinfeld sokolov type hamiltonian symmetry reduction applied to a poisson submanifold in the dual of the lie algebra widehat gl pr 1 here a series of extensions of this matrix gelfand dickey system is derived by means of a generalized drinfeld sokolov reduction defined for the lie algebra widehat gl pr s 1 using the natural embeddinggl pr glpr s fors any positive integer the hierarchies obtained admit a description in terms of ap p matrix pseudo differential operator comprising anr kdv type positive part and a non trivial negative part this system has been investigated previously in thep 1 case as a constrained kp system in this paper the previous results are considerably extended and a systematic study is presented on the basis of the drinfeld sokolov approach that has the advantage that it leads to local poisson brackets and makes clear the conformal w algebra structures related to the kdv type hierarchies

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

tau functions and generalized integrable hierarchies the tau function formalism for a class of generalized zero curvature integrable hierarchies of partial differential equations is constructed the class includes the drinfel d sokolov hierarchies a direct relation between the variables of the zero curvature formalism and the tau functions is established the formalism also clarifies the connection between the zero curvature hierarchies and the hirotatype hierarchies of kac and wakimoto

the quadratic hamiltonian theorem in infinite dimensions it is shown that a smooth diffeomorphism on a symplectic banach space is canonical if it is canonoid with respect to all dynamical systems whose hamiltonian functions are quadratic

frobenius manifolds and central invariants for the drinfeld sokolov bihamiltonian structures the drinfeld sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine lie algebra we compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of miura type transformations

equivalence of two approaches to integrable hierarchies of kdv type the equivalence between the approaches of drinfeld sokolov and feigin frenkel to the mkdv and kdv hierarchies is established a new derivation of the mkdv equations in the zero curvature form is given connection with the baker akhiezer function and the tau function is also discussed

a generalized wigner function on the space of irreducible representations of the weyl heisenberg group and its transformation properties a natural extension of the wigner function to the space of irreducible unitary representations of the weyl heisenberg group is discussed the action of the automorphisms group of the weyl heisenberg group onto wigner functions and their generalizations and onto symplectic tomograms is elucidated some examples of physical systems are considered to illustrate some aspects of the characterization of the wigner functions as solutions of differential equations

hamiltonian structures of a coupled ramani equation abstract with the aid of hamiltonian structures for the constrained modified kp flow we show that a coupled ramani equation is a bi hamiltonian system with a local hamiltonian operator p 2 and a symplectic operator after a reduction we recover the bi hamiltonian formalism of the ramani equation furthermore we obtain the explicit bi hamiltonian structure for the drinfeld sokolov hierarchies of c 3 1 using the local hamiltonian operator p 2

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

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

multidimensional exponential timestepping with boundary test exponential timestepping algorithms are efficient for exit time problems because a boundary test can be performed at the end of each timestep giving high order convergence in numerical evaluation of mean exit times successive time increments are independent random variables with an exponential distribution we show how to perform exact timestepping for brownian motion in more than one dimension and consider hitting times of curved surfaces taking as examples the exit times from a circle from a ball from an ellipse and from the region bounded by the two lines of a hyperbola we report the results of numerical experiments performed with the algorithms developed in this work

alternative structures and bi hamiltonian systems on a hilbert space we discuss transformations generated by dynamical quantum systems which are bi unitary i e unitary with respect to a pair of hermitian structures on an infinite dimensional complex hilbert space we introduce the notion of hermitian structures in generic relative position we provide a few necessary and sufficient conditions for two hermitian structures to be in generic relative position to better illustrate the relevance of this notion the group of bi unitary transformations is considered in both the generic and the non generic case finally we generalize the analysis to real hilbert spaces and extend to infinite dimensions results already available in the framework of finite dimensional linear bi hamiltonian systems

additional symmetries of generalized integrable hierarchies the non isospectral symmetries of a general class of integrable hierarchies are found by generalizing the galilean and scaling symmetries of the korteweg de vries equation and its hierarchy the symmetries arise in a very natural way from the semi direct product structure of the virasoro algebra and the affine kac moody algebra underlying the construction of the hierarchy in particular the generators of the symmetries are shown to satisfy a subalgebra of the virasoro algebra when a tau function formalism is available the infinitesimal symmetries act directly on the tau functions as moments of virasoro currents some comments are made regarding the role of the non isospectral symmetries and the form of the string equations in matrix model formulations of quantum gravity in two dimensions and related systems

from additional symmetries to linearization of virasoro symmetries abstract we construct the additional symmetries and derive the adler shiota van moerbeke formula for the two component bkp hierarchy considered as certain reductions of the two component bkp hierarchy the drinfeld sokolov hierarchies of type d are proved to possess symmetries written as the linear action of a series of virasoro operators on the tau function it results in that the drinfeld sokolov hierarchies of type d coincide with dubrovin and zhang s hierarchies associated to the frobenius manifolds for coxeter groups of type d and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the virasoro operators and the time derivations of the hierarchy

alternative structures and bi hamiltonian systems in the study of bi hamiltonian systems both classical and quantum one starts with a given dynamics and looks for all alternative hamiltonian descriptions it admits in this paper we start with two compatible hermitian structures the quantum analogue of two compatible classical poisson brackets and look for all the dynamical systems which turn out to be bi hamiltonian with respect to them

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 witten equation mirror symmetry and quantum singularity theory for any nondegenerate quasi homogeneous hypersurface singularity we describe a family of moduli spaces a virtual cycle and a corresponding cohomological eld theory associated to the singularity this theory is analogous to gromov witten theory and generalizes the theory of r spin curves which corresponds to the simple singularity ar 1 we also resolve two outstanding conjectures of witten the rst conjecture is that ade singularities are self dual and the second conjecture is that the total potential functions of ade singularities satisfy corresponding ade integrable hierarchies other cases of integrable hierarchies are also discussed

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

non symplectic symmetries and bi hamiltonian structures of the rational harmonic oscillator the existence of bi hamiltonian structures for the rational harmonic oscillator non central harmonic oscillator with rational ratio of frequencies is analysed by making use of the geometric theory of symmetries we prove that these additional structures are a consequence of the existence of dynamical symmetries of non symplectic non canonical type the associated recursion operators are also obtained

drinfeld sokolov hierarchies and diagram automorphisms of affine kac moody algebras for a diagram automorphism of an affine kac moody algebra such that the folded diagram is still an affine dynkin diagram we show that the associated drinfeld sokolov hierarchy also admits an induced automorphism then we show how to obtain the drinfeld sokolov hierarchy associated to the affine kac moody algebra that corresponds to the folded dynkin diagram from the invariant sub hierarchy of the original drinfeld sokolov hierarchy

bihamiltonian cohomologies and integrable hierarchies ii the tau structures starting from a so called flat exact semisimple bihamiltonian structure of hydrodynamic type we arrive at a frobenius manifold structure and a tau structure for the associated principal hierarchy we then classify the deformations of the principal hierarchy which possess tau structures