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

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

recursion relations for unitary integrals combinatorics and the toeplitz lattice in a discussion in spring 2001 alexei borodin showed us recursion relations for the toeplitz determinants going with the symbols e t z z 1 and 1 xiz alpha 1 xiz 1 beta borodin obtained these relations using riemann hilbert methods see the recent work of borodin b and baik baik the nature of borodin s recursion relations pointed towards the toeplitz lattice and its virasoro algebra introduced by us in 3 in this paper we take the toeplitz lattice and virasoro algebra approach for a fairly large class of symbols leading to a systematic way of generating recursion relations the latter are very naturally expressed in terms of the l matrices appearing in the toeplitz lattice equations as a surprise we find compared to borodin s a different set of relations except for the 3 step relations associated with the symbol e t z z 1 the painleve analysis of the toeplitz lattice enables us to show the singularity confinement for these recursion relations

the r matrix theory and the reduction of poisson manifolds the relation between the r matrix approach to the construction of bi hamiltonian structures and the reduction of poisson manifolds is discussed the crucial role of the reduction process is emphasized by showing on the basis of specific examples that the knowledge of a r matrix may not be sufficient the reduction of the poisson structures related with r from the entire algebra to the lax submanifold may become an obstruction in constructing the bi hamiltonian structures of a specific hierarchy of integrable equations

peakons r matrix and toda lattice abstract the integrability of a family of hamiltonian systems describing in a particular case the motion of n peakons special solutions of the so called camassa holm equation is established in the framework of the r matrix approach starting from its lax representation in the general case the r matrix is a dynamical one and has an interesting though complicated structure however for a particular choice of the relevant parameters in the hamiltonian the one corresponding to the pure peakons case the r matrix becomes essentially constant and reduces to the one pertaining to the finite non periodic toda lattice intriguing consequences of this property are discussed and an integrable time discretisation is derived

mastersymmetries angle variables and recursion operator of the relativistic toda lattice conserved quantities bi hamiltonian formulation and recursive structure of the relativistic toda lattice rt are obtained in an algorithmic way without making use of the lax representation furthermore for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries a new hierarchy of completely integrable systems is discovered which turns out to correspond to the negative of the hierarchy of rt thus it is shown that the full algebra of time dependent symmetry group generators for each member of the rt hierarchy is isomorphic to the algebra of first order differential operators with laurent polynomials as coefficients the surprising phenomenon is revealed that the members of the rt hierarchy are connected to their negative counterparts by explicit backlund transformation

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

poisson structures of calogero moser and ruijsenaars schneider models we examine the hamiltonian structures of some calogero moser and ruijsenaars schneider n body integrable models we propose explicit formulations of the bi hamiltonian structures for the discrete models and field theoretical realizations of these structures we discuss the relevance of these realizations as collective field theory for the discrete models

on deformations of standard r matrices for integrable infinite dimensional systems simple deformations with a parameter of classical r matrices which follow from decomposition of appropriate lie algebras are considered as a result nonstandard lax representations for some well known integrable systems are presented and integrable evolution equations are constructed

characteristic methods for multidimensional evolution equations abstract the stochastic method of characteristics is generalized to analyze multidimensional evolution equations of arbitrary order with constant coefficients a linear space of abstract characteristics is introduced along with algorithms for associating characteristics with evolutions a projection from a space of functions of characteristics to solutions of evolution equations plays the role of a generalized expectation the solution is then viewed as an average of initial data examples are presented which indicate possible extensions of the theory presented here to equations with variable coefficients

the hamiltonian structure of the nonabelian toda hierarchy we show that a subset of the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger operator is endowed with a hamiltonian structure and exhibits an infinite number of integrals of motion in involution we also establish the relation between these integrals of motion and the transmission coefficient of the underlying linear problem and show that such a relation implies that for the whole class previously introduced there exists an infinite number of conserved quantities

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

nijenhuis g manifolds and lenard bicomplexes a new approach to kp systems we suggest a method to extend the theory of recursion operators to integrable hamiltonian systems in two space dimensions like kp systems the approach aims to stress the conceptual unity of the theories in one and two space dimensions a sound explanation of the appearance of bilocal operators is also given

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

incomplete sampling series and the recovery of missing samples from oversampled band limited signals it is well known that a band limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost it is shown that an alternative simple proof of this fact carries over to generalized sampling expansions more precisely it is shown that any finite number of missing samples can be recovered from the remaining ones in the case of generalized kramer sampling expansions if an appropriate oversampling constraint is satisfied the recovery can be accomplished either iteratively or noniteratively

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

fundamentals of higher order stochastic equations stochastic processes that correspond to equations of arbitrary differential order are obtained we show that stochastic paths in a complex extension of the original phase space allow an implementation of such higher order derivative terms the resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments however the correspondence has unusual properties only the analytic moments are convergent while non analytic moments such as the complex moduli are non convergent these results unify previous approaches that transform higher derivative equations into probabilistic stochastic equations larger ensembles are required as time steps are reduced giving these equations unusual convergence properties this type of process is relevant to the question of how to obtain stochastic quantum simulations using the wigner representation

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

a two component generalization of the degasperis procesi equation we present two different hamiltonian extensions of the degasperis procesi equation to two component equations the construction is based on the observation that the second hamiltonian operator of the degasperis procesi equation could be considered as the dirac reduced poisson tensor of the second hamiltonian operator of the boussinesq equation the first extension is generated by the hamiltonian operator which is a dirac reduced operator of the generalized but degenerated second hamiltonian operator of the boussinesq equation the second one is obtained by the n 2 supersymmetric extension of the aforementioned method as the byproduct of this procedure we obtain the hamiltonian system of interacting equations which contains the camassa holm and degasperis procesi equations

backlund transformations for the kp and mkp hierarchies with self consistent sources using gauge transformations for the corresponding generating pseudo differential operators ln in terms of eigenfunctions and adjoint eigenfunctions we construct several types of auto backlund transformations for the kp hierarchy with self consistent sources kphscs and mkp hierarchy with self consistent sources mkphscs respectively the backlund transformations from kphscs to mkphscs are also constructed in this way

asymmetric multi channel sampling in shift invariant spaces abstract we develop an asymmetric multi channel sampling on a shift invariant space v with a riesz generator t in l 2 r where each channeled signal is assigned a uniform but distinct sampling rate we use fourier duality between v and l 2 0 2 to find conditions under which there is a stable asymmetric multi channel sampling formula on v

recovery of missing samples for oversampling in shift invariant spaces abstract it is well known that any finite missing samples of a band limited signal can be recovered from the remaining known samples when the signal is oversampled at a rate higher than the minimum nyquist rate we consider a similar problem for oversampled signals in shift invariant spaces with compactly supported continuous riesz generators using a multi channel oversampling we find conditions under which finitely many missing samples can be recovered

an inverse scattering formalism for higher order differential operators abstract an inverse scattering formalism is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the gel fand levitan theory for sturm liouville operators under suitable hypotheses it is proved that for given n 2 the coefficients in these operators are uniquely determined by a single spectral matrix our proof makes use of a transformation due to m k fage which generalizes the povzner levitan transformation for sturm liouville operators

generalization of the diffusion equation by using the maximum entropy principle by using the so called maximum entropy principle in information theory one derives a generalization of the fokker planck kolmogorov equation which applies when the n first transition moments of the process are proportional to t while the other ones can be neglected

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

regular flat structure and generalized okubo system we study a relationship between regular flat structures and generalized okubo systems we show that the space of variables of isomonodromic deformations of a regular generalized okubo system can be equipped with a flat structure as its consequence we introduce flat structures on the spaces of independent variables of generic solutions to classical painleve equations except for pi in our framework the painleve equations pvi pii can be treated uniformly as just one system of differential equations called the four dimensional extended wdvv equation then the well known coalescence cascade of the painleve equations corresponds to the degeneration scheme of the jordan normal forms of a square matrix of rank four

4 dimensional frobenius manifolds and painleve vi a frobenius manifold has tri hamiltonian structure if it is even dimensional and its spectrum is maximally degenerate we study the case of the lowest nontrivial dimension n 4 and show that under the assumption of semisimplicity the corresponding isomonodromic fuchsian system is described by the painleve hbox vi mu equation since the solutions of this equation are known to parametrize semisimple frobenius manifolds of dimension n 3 this leads to an explicit procedure mapping 3 dimensional frobenius structures of 4 dimensional ones and giving all tri hamiltonian structures in four dimensions we illustrate the construction by computing two examples in the framework of frobenius structures on hurwitz spaces

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

local and non local multiplicative poisson vertex algebras and differential difference equations we develop the notions of multiplicative lie conformal and poisson vertex algebras local and non local and their connections to the theory of integrable differential difference hamiltonian equations we establish relations of these notions to q deformed w algebras and lattice poisson algebras we introduce the notion of adler type pseudodifference operators and apply them to integrability of differential difference hamiltonian equations

multi hamiltonian structures for a class of degenerate completely integrable systems in this paper a class of degenerate i e associated to a degenerate poisson structure completely integrable systems is studied which generalizes the so called odd and even master systems introduced and studied by mumford and by vanhaecke it is shown that all these completely integrable systems called the generalized master systems admit a multi hamiltonian formulation and a systematic construction of this multi hamiltonian structure is described

lax nijenhuis operators for integrable systems the relationship between lax and bi hamiltonian formulations of dynamical systems on finite or infinite dimensional phase spaces is investigated the lax nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the lenard recursion relations while the converse holds for an operator with a simple spectrum explicit higher order hamiltonian structures for the toda system a second hamiltonian structure of the euler equation for a rigid body in n dimensional space and the quadratic adler gelfand dickey structure for the kdv hierarchy are derived using the lax nijenhuis equation

construction of a complex valued fractional brownian motion of order n in this paper a brownian motion of order n n 2 is defined by a probabilistic approach different from hochberg s and mandelbrot s this process is constructed from sums of independent r 1 n valued random variables rv where r 1 n z c zn r many properties of the real standard brownian motion are generalized at order n but in the case n 2 it is interesting to describe the brownian motion of order n on the algebra b r 1 n r where b r 1 n is the algebra generated by sets of type a 0 h z c zn 0 hn her this algebra is totally different from b r r thus this study shows the fractal nature of the brownian motion of order n and given invariance scale self similarity properties then a stochastic integral and an ito taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average all these results can be obtained via nonstandard analysis methods infinitesimal time discretization finally one rem

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

r structures yang baxter equations and related involution theorems there are several ways to construct functions in involution on a lie bi algebra a lie algebra equipped with a second lie bracket for the solvable systems associated to the casimir functions a second hamiltonian formulation can be constructed and a class of bi hamiltonian korteweg de vries like evolution equations with explicit space dependence is derived translating the casimir functions with the flow of a special vector field yields another set of functions in involution lenard relations are found for the corresponding hamiltonian systems finally solutions of the classical yang baxter equation lead to an analog of compatible hamiltonian pairs the invariants of the resulting hereditary operators are in involution

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

nonlocal quadratic poisson algebras monodromy map and bogoyavlensky lattices a new lax representation for the bogoyavlensky lattice is found and its r matrix interpretation is elaborated the r matrix structure turns out to be related to a highly nonlocal quadratic poisson structure on a direct sum of associative algebras the theory of such nonlocal structures is developed and the poisson property of the monodromy map is worked out in the most general situation some problems concerning the duality of lax representations are raised

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

explicit examples of hurwitz frobenius manifolds in genus one we provide three new solutions to the witten dijkgraaf verlinde verlinde wdvv equations by computing examples of prepotentials of hurwitz frobenius manifolds their deformations and real doubles by applying the general formalism to two families of elliptic curves we obtain solutions to the wdvv system depending on four and eight variables we also correct mistakes in previously known solutions in dimensions four and six associated with the same families of curves we provide three new solutions to the witten dijkgraaf verlinde verlinde wdvv equations by computing examples of prepotentials of hurwitz frobenius manifolds their deformations and real doubles by applying the general formalism to two families of elliptic curves we obtain solutions to the wdvv system depending on four and eight variables we also correct mistakes in previously known solutions in dimensions four and six associated with the same families of curves

central limit theorem for products of random matrices using the semigroup product formula of p chernoff a central limit theorem is derived for products of random matrices applications are presented for representations of solutions to linear systems of stochastic differential equations and to the corresponding partial differential evolution equations included is a discussion of stochastic semigroups and a stochastic version of the lie trotter product formula

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes

product formulas for solutions of initial value partial differential equations i abstract families of operators which approximate semi groups or evolution systems generated by partial differential operators are constructed product formulas are used to recover these semi groups or evolution systems through product integrals conditions on generators are provided under which its semi group or evolution system can be approximated in this way by families of specific types of operators