canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

classical limits of unbounded quantities by strict quantization this paper extends the tools of c algebraic strict quantization toward analyzing the classical limits of unbounded quantities in quantum theories we introduce the approach first in the simple case of finite systems then we apply this approach to analyze the classical limits of unbounded quantities in bosonic quantum field theories with particular attention to number operators and hamiltonians the methods take classical limits in a representation independent manner and so allow one to compare quantities appearing in inequivalent fock space representations

rich classes inferable from positive data inductive inference from positive data is shown to be remarkably powerful using the framework of elementary formal systems an elementary formal system efs for short is a kind of logic program on consisting of finitely many axioms any context sensitive language is definable by a restricted efs called a length bounded efs length bounded efss with at most n axioms are considered and it is shown that inductive inference from positive data works successfully for their models as well as for their languages from this it follows that any class of logic programs such as prolog programs corresponding to length bounded efss can be inferred from positive facts

infrared singular fields and nonregular representations of canonical commutation relation algebras infrared singular variables which often enter in the formulation of models in quantum field theory many body theory and quantum statistical mechanics are described in terms of nonregular representations of canonical commutation relations algebras general properties of such representations are discussed in this way one also obtains insight into the construction and representation of charged field algebras a as canonical extensions of a neutral observable algebra a0 the charged state representations of a0 are obtained through nonregular vacuum representations of the extended field algebra a the above structures are proven to work in several explicit models

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

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

construction and uniqueness of the c weyl algebra over a general pre symplectic space a systematic approach to the c weyl algebra w e over a possibly degenerate pre symplectic form on a real vector space e of possibly infinite dimension is elaborated in an almost self contained manner the construction is based on the theory of kolmogorov decompositions for positive definite functions on involutive semigroups and their associated projective unitary group representations the positive definite functions provide also the c norm of w e the latter being shown to be isomorphic to the twisted group c algebra of the discrete vector group e the connections to related constructions are indicated the treatment of the fundamental symmetries is outlined for arbitrary pre symplectic the construction method is especially applied to the trivial symplectic form 0 leading to the commutative weyl algebra over e which is shown to be isomorphic to the c algebra of the almost periodic continuous function on the topological dual e of e with respect to an arbitrary locally convex ha

iterative learning from positive data and negative counterexamples a model for learning in the limit is defined where a so called iterative learner gets all positive examples from the target language tests every new conjecture with a teacher oracle if it is a subset of the target language and if it is not then it receives a negative counterexample and uses only limited long term memory incorporated in conjectures three variants of this model are compared when a learner receives least negative counterexamples the ones whose size is bounded by the maximum size of input seen so far and arbitrary ones a surprising result is that sometimes absence of bounded counterexamples can help an iterative learner whereas arbitrary counterexamples are useless we also compare our learnability model with other relevant models of learnability in the limit study how our model works for indexed classes of recursive languages and show that learners in our model can work in non u shaped way never abandoning the first right conjecture

language learning from texts mindchanges limited memory and monotonicity abstract the paper explores language learning in the limit under various constraints on the number of mindchanges memory and monotonicity we define language learning with limited long term memory and prove that learning with limited memory is exactly the same as learning via set driven machines when the order of the input string is not taken into account further we show that every language learnable via a set driven machine is learnable via a conservative machine making only justifiable mindchanges we get a variety of separation results for learning with bounded number of mindchanges or limited memory under restrictions on monotonicity a surprising result is that there are families of languages that can be monotonically learned with at most one mindchange but can neither be weak monotonically nor conservatively learned many separation results have a variant if a criterion a can be separated from b then often it is possible to find a family l of languages such that l is a and b learnable but while it is possible to restrict the number of mindchanges or long term memory on criterion a this is impossible for b

quantum harmonic analysis on phase space relative to an irreducible representation of the canonical commutation relations convolutions between quantum mechanical operators and between functions and operators are defined for which the usual weyl transform acts as a fourier transform basic properties of these operations are developed in close analogy to harmonic analysis on r2n using the quantum version of wiener s approximation theorem a natural one to one correspondence between the closed phase space translation invariant subspaces of classical and quantum observables is established

the integrable hierarchy constructed from a pair of kdv type hierarchies and its associated w algebra for any two arbitrary positive integers n and m using them th kdv hierarchy and the n m th kdv hierarchy as building blocks we are able to construct another integrable hierarchy referred to as the n m th kdv hierarchy thew algebra associated to the second hamiltonian structure of the n m th kdv hierarchy calledw n m algebra is isomorphic via a miura map to the direct sum of aw m algebra aw n m algebra and an additionalu 1 current algebra in turn from the latter we can always construct a representation of aw algebra

string theory and water waves we uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non perturbatively we are able to embed the type 0a and 0b a a minimal string theories into this single framework the string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy we observe that there are several other string like limits of the system and conjecture that some of them are type iia and iib a d minimal string backgrounds we explain how these and several string like special points arise and are connected in some cases the framework endows the theories with a non perturbative definition for the first time notably we discover that the painlev iv equation plays a key role in organizing the string theory physics joining its siblings painlev i and ii whose roles have previously been identified in this minimal string context

toeplitz operators and quantum mechanics abstract toeplitz operators on the segal bargmann spaces of gaussian measure square integrable entire functions on complex n space c n are studied the c algebra generated by the weyl form of the canonical commutation relations consists precisely of the uniform limits of almost periodic toeplitz operators the question of which toeplitz operators admit a symbol calculus modulo the compact operators is raised and sufficient conditions are given for such a calculus these conditions involve a notion of slow oscillation at infinity

the hamiltonian structures of the nonlinear schrodinger equation in the classical limit using madelung s hydrodynamical variables it is shown that the bi hamiltonian structure of the nonlinear schrodinger equation goes over to a bi hamiltonian structure of the shallow water wave equations in the classical limit

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

regular articlesquared eigenfunction symmetries for soliton equations part ii nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

an example of a miura map the shallow water wave equations the linear hamiltonian structure of modified shallow water wave equations goes to the non linear and second hamiltonian structure of the bi hamiltonian shallow water wave equations a similar situation must exist for gas dynamics according to a result of dubrovin and novikov 1983

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

comparison of identification criteria for machine inductive inference abstract a natural pl 1 hierarchy of successively more general criteria of success for inductive inference machines is described based on the size of sets of anomalies in programs synthesized by such machines these criteria are compared to others in the literature some of our results are interpreted as tradeoff results or as showing the inherent relative computational complexity of certain processes and others are interpreted from a positivistic mechanistic philosophical stance as theorems in philosophy of science the techniques of recursive function theory are employed including ordinary and infinitary recursion theorems

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

the classical limit of a state on the weyl algebra this paper considers states on the weyl algebra of the canonical commutation relations over the phase space r 2n we show that a state is regular iff its classical limit is a countably additive borel probability measure on r 2n it follows that one can reduce the state space of the weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

learning languages from positive data and a finite number of queries a computational model for learning languages in the limit from full positive data and a bounded number of queries to the teacher oracle is introduced and explored equivalence superset and subset queries are considered for the latter one we consider also a variant when the learner tests every conjecture but the number of negative answers is uniformly bounded if the answer is negative the teacher may provide a counterexample we consider several types of counterexamples arbitrary least counterexamples the ones whose size is bounded by the size of positive data seen so far and no counterexamples a number of hierarchies based on the number of queries answers and types of answers counterexamples is established capabilities of learning with different types of queries are compared in most cases one or two queries of one type can sometimes do more than any bounded number of queries of another type still surprisingly a finite number of subset queries is sufficient to simulate the same number of equivalence queries when behaviourally correct learners do not receive counterexamples and may have unbounded number of errors in almost all conjectures

bifurcation of travelling wave solutions for the modified dispersive water wave equation abstract the method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation we obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits in different regions different types of travelling solutions including solitary wave solutions shock wave solutions and periodic wave solutions are simulated furthermore with a generalized projective riccati equation method several new explicit exact solutions are obtained

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

the resolvent algebra a new approach to canonical quantum systems the standard c algebraic version of the algebra of canonical commutation relations the weyl algebra frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does it incorporate pertinent physical observables such as bounded functions of the hamiltonian here a novel c algebra of the canonical commutation relations is presented which does not suffer from such problems it is based on the resolvents of the canonical operators and their algebraic relations the resulting c algebra the resolvent algebra is shown to have many desirable analytic properties and the regularity structure of its representations is surprisingly simple moreover the resolvent algebra is a convenient framework for applications to interacting and to constrained quantum systems as we demonstrate by several examples