on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

bi integrable couplings of a new soliton hierarchy associated with a non semisimple lie algebra we construct bi integrable couplings for a new soliton hierarchy from zero curvature equations associated with a non semisimple loop algebra we demonstrate the liouville integrability of the hierarchy by showing that the bi integrable couplings generated are bi hamiltonian

iterated rings of bounded elements and generalizations of schmudgen s positivstellensatz let a be a commutative r algebra of finite transcendence degree d 2 n we investigate the relationship between the subring of geometrically bounded elements

strict positivstellensatze for matrix polynomials with scalar constraints abstract we extend krivine s strict positivstellensatz for usual real multivariate polynomials to symmetric matrix polynomials with scalar constraints the proof is an elementary computation with schur complements analogous extensions of schmudgen s and putinar s strict positivstellensatz were recently proved by hol and scherer using methods from optimization theory

a trace identity and its application to integrable systems of 1 2 dimensions a trace identity is proposed and it is shown that this identity can be effectively applied to establish the hamiltonian structure of the kp ds hierarchies and a new hierarchy of 1 2 dimensional systems

two symmetry constraints for a generalized dirac integrable hierarchy abstract by a two by two matrix spectral problem a generalized dirac integrable hierarchy is presented a hamiltonian structure of the obtained hierarchy is established by trace identity and its liouville integrability is proved then through bargmann symmetry constraint spatial part of the lax pairs and adjoint lax pairs is nonlinearized as a completely integrable finite dimensional hamiltonian system next under an implicit symmetry constraint both spatial part and temporal parts of the lax pairs and adjoint lax pairs are all nonlinearized as completely integrable finite dimensional hamiltonian systems ultimately the involutive representation of solution of the generalized dirac integrable hierarchy is given

an algebro geometric solution for a hamiltonian system with application to dispersive long wave equation by using an iterative algebraic method we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation it is shown that the hierarchy is integrable in liouville sense and possesses bi hamiltonian structure two commutators with zero curvature and lax representations for the hierarchy are constructed respectively by using two different systematic methods under a bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional hamiltonian system by introducing the abel jacobi coordinates an algebro geometric solution for the dispersive long wave equation is derived by resorting to the riemann theta function

positive polynomials and sequential closures of quadratic modules let s x n f 1 x 0 f s x 0 be a basic closed semi algebraic set in n and let po f 1 f s be the corresponding preordering in x 1 x n we examine for which polynomials f there exist identities f eq po f 1 f s for all e 0 these are precisely the elements of the sequential closure of po f 1 f s with respect to the finest locally convex topology we solve the open problem from kuhlmann marshall and schwartz 2002 2005 whether this equals the double dual cone po f 1 f s vv by providing a counterexample we then prove a theorem that allows us to obtain identities for polynomials as above by looking at a family of fibre preorderings constructed from bounded polynomials these fibre preorderings are easier to deal with than the original preordering in general for a large class of examples we are thus able to show that either every polynomial f that is nonnegative on s admits such representations or at least the polynomials from po f 1 fs vv do the results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings

a hierarchy of nonlinear evolution equations its bi hamiltonian structure and finite dimensional integrable systems an isospectral problem and the associated hierarchy of nonlinear evolution equations is presented as a reduction a new generalized nonlinear schrodinger equation is obtained it is shown that the hierarchy possesses bi hamiltonian structure and is integrable in liouville sense moreover the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenvalues

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

some new integrable nonlinear evolution equations and darboux transformation two classes of new nonlinear evolution equations are derived with the aid of a 4 4 matrix spectral problem with three potentials a darboux transformation for the first nontrivial nonlinear evolution equation is given based on the gauge transformation between the corresponding 4 4 matrix spectral problems as an application of the darboux transformation some explicit solutions of the nontrivially nonlinear evolution equation are obtained

a hierarchy of coupled burgers systems possessing a hereditary structure a hierarchy of typical integrable coupled burgers systems is proposed by introducing a special spectral problem involving q dependent variables u0 u1 uq 1 these systems possess a hereditary structure and may reduce to a hierarchy of burgers equations under the reduction ui 0 0 or i or q 2 it is shown that their flows and lax operators commute mutually but each system is not hamiltonian

a hierarchy of systems of non linear equations imposing isospectral invariance for the one dimensional dirac operator yields an infinite hierarchy of systems of chiral invariant non linear partial differential equations the same system is obtained through a lax pair construction and finally a formulation in terms of kac moody generators is given

integrable counterparts of the d kaup newell soliton hierarchy two integrable counterparts of the d kaup newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal lie algebra so 3 r an application of the trace identity presents hamiltonian or quasi hamiltonian structures of the resulting counterpart soliton hierarchies thereby showing their liouville integrability i e the existence of infinitely many commuting symmetries and conserved densities the involved hamiltonian and quasi hamiltonian properties are shown by computer algebra systems

the generalized fractional trace variational identity and fractional integrable couplings of kaup newell hierarchy starting from fractional isospectral problems and general bilinear forms the generalized fractional trace identity is presented then a new explicit lie algebra is introduced for which the fractional integrable couplings of kaup newell hierarchy are derived furthermore we obtained the fractional hamiltonian structures of the fractional integrable couplings of kaup newell hierarchy the methods derived by us can be generalized to other fractional integrable couplings

nonlinear continuous integrable hamiltonian couplings based on a kind of special non semisimple lie algebras a scheme is presented for constructing nonlinear continuous integrable couplings variational identities over the corresponding loop algebras are used to furnish hamiltonian structures for the resulting continuous integrable couplings the application of the scheme is illustrated by an example of nonlinear continuous integrable hamiltonian couplings of the akns hierarchy of soliton equations

darboux transformation and hamiltonian structure for the jaulent miodek hierarchy abstract starting from the jaulent miodek spectral problem we derive the associated hierarchy of nonlinear evolution equations in this paper it is shown that this hierarchy is completely integrable in the liouville sense and possesses the hamiltonian structure moreover by virtue of symbolic computation two types of darboux transformations for the whole hierarchy are explicitly constructed which enables us to find the new soliton like shock and anti shock solutions for the jaulent miodek hierarchy figures are presented to discuss the properties of the new soliton like solutions

some results on the sarkovskii partial ordering of permutations if 7 is a cyclic permutation and x is a periodic point of a continuous function f r i r with period x order 7r n then we say that x has type 7 if the orbit of x consists of points xl x2 xn with f xi x i i in analogy with sarkovskii s theorem we define a partial ordering on cyclic permutations by 0 7x if every continuous function with a periodic point of type 0 also has a point of type 7 in this paper we examine this partial order form the point of view of critical points itineraries and kneading sequences we show that 0 xr if and only if the maxima of 0 are higher and the minima lower than those of 7 where higher and lower are precisely defined in terms of itineraries we use this to obtain several results about x there are no minimal upper bounds if 7 and 0 have the same number of critical points or if they differ by 1 or sometimes 2 then 0 xr if and only if 0 7r for some period double 7r of 7r and finally we prove a conjecture of baldwin that maximal permutations of size n have n 2 critical points and obtain necessary and sufficient conditions for such a permutation to be maximal

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

a spectral problem based on so 3 r and its associated commuting soliton equations we present a spectral problem based on the three dimensional real special orthogonal lie algebra so 3 r and construct a hierarchy of commuting bi hamiltonian soliton equations by zero curvature equations associated with the spectral problem an illustrative example of soliton equations is computed together with its associated bi hamiltonian structure

two integrable hamiltonian hierarchies in sl 2 r and so 3 r with three potentials by introducing two specific matrix spectral problems associated with sl 2 r and so 3 r matrix lie algebras we generate two integrable hamiltonian hierarchies with three potentials the computation and analysis on their hamiltonian structures by means of the trace identity show that the resulting hierarchies are liouville integrable namely that each hierarchy consists of commuting hamiltonian soliton equations

a soliton hierarchy associated with a new spectral problem and its hamiltonian structure a hierarchy of soliton equations together with its hamiltonian structure is constructed from a new spectral problem associated with the three dimensional special orthogonal real lie algebra so 3 the liouville integrability of the presented soliton hierarchy is proved based on the hamiltonian structure

coupled camassa holm equations n peakons and infinitely many conservation laws abstract three coupled camassa holm equations are proposed which are associated with a 4 4 matrix spectral problem with two potentials with the aid of the zero curvature equation we derive a hierarchy of new nonlinear evolution equations and establish their bi hamiltonian structures the three coupled camassa holm equations are exactly negative flows in the hierarchy and exact solutions with n peakons and infinite sequences of conserved quantities are discussed

new hierarchies of isospectral and non isospectral integrable nlees derived from the harry dym spectral problem abstract by making use of our previous framework we display the hierarchies of generalized nonlinear evolution equations gnlees associated with the harry dym spectral problem and construct the corresponding generalized lax representations glr in this paper it will be clearly seen that the generalized hierarchies can include not only the well known harry dym hierarchy of isospectral nlees but also other new integrable hierarchies of isospectral and non isospectral nlees through choosing the so called seed function these new hierarchies of nlees give some new integrable evolution equations which are very likely to have potential applications in theoretical and experimental physics all of these hierarchies of nlees possess the glr

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

a hierarchy of non linear evolution equations its hamiltonian structure and classical integrable system abstract an isospectral problem and a new corresponding hierarchy of non linear evolution equations are proposed as a reduction a new generalized non linear schrodinger equation is obtained it is shown that the hierarchy of equations possesses a bi hamiltonian structure the eigenvalue problem is non linearized so as to be a finite dimensional completely integrable system under a constraint between the potentials and the eigenfunctions

topological conjugacy and transitivity for a class of piecewise monotone maps of the interval on dit qu une application continue f d un intervalle compact dans lui meme est markov lineaire s il est lineaire par morceaux et l ensemble de tous les f k x ou k 0 et x une extremite d un morceau lineaire est fini on donne une classification effective pour des applications de markov lineaires et une procedure effective pour determiner quand une telle application est transitive

using factorization to solve soliton equations it was shown in our previous paper that each equation in a soliton hierarchy can be factorized into two commuting x and finite dimensional integrable hamiltonian systems fdihss the separation variables for these fdihss are constructed by using their lax representation by means of the factorization and the separability of the fdihss we obtain the jacobi inversion problem which is solvable in terms of riemann theta functions for soliton equations this provides a method analogous to the separation of variables for solving soliton equations

a similar extension of the wki hierarchy its hamiltonian structure and the associated finite dimensional hamiltonian system abstract an isospectral problem and a similar extension of the wadati konno ichikawa hierarchy are proposed by using the trace identity it is shown that the hierarchy of equations possesses a bi hamiltonian structure the eigenvalue problem is nonlinearized to a finite dimensional hamiltonian system under a constraint between the potentials and the eigenfunctions

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

application of jacobi s representation theorem to locally multiplicatively convex topological real algebras abstract let a be a commutative unital r algebra and let be a seminorm on a which satisfies a b a b we apply t jacobi s representation theorem 10 to determine the closure of a a 2 d module s of a in the topology induced by for any integer d 1 we show that this closure is exactly the set of all elements a a such that a 0 for every continuous r algebra homomorphism a r with s 0 and that this result continues to hold when is replaced by any locally multiplicatively convex topology on a we obtain a representation of any linear functional l a r which is continuous with respect to any such or and nonnegative on s as integration with respect to a unique radon measure on the space of all real valued r algebra homomorphisms on a and we characterize the support of the measure obtained in this way

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems i the algebraic structure of the gradient holonomic algorithm for lax integrable dynamical systems is discussed a generalization of the r structure approach for the case of operator valued affine lie algebras is used to prove the bi hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions the monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical lie poisson bracket on initial affine lie algebra with gauge central extension as an illustration the two dimensional operator benney kaup integrable hierarchy is considered and their bi hamiltonicity is proved

nonlinear evolution equations connected with the matrix schrodinger spectral problem a generalized 2 2 schrodinger spectral problem and n n reduced matrix schrodinger spectral problem are considered the corresponding hierarchies of nonlinear evolution equations are derived

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

bi integrable couplings and tri integrable couplings of the modified ablowitz kaup newell segur hierarchy with self consistent sources we discuss the bi integrable couplings and tri integrable couplings of the modified ablowitz kaup newell segur akns hierarchy with self consistent sources from the enlarged matrix spectral problem and the enlarged zero curvature equations by using the variational identities over non semisimple lie algebras consisting of block matrices we construct hamiltonian formulations for integrable couplings bi integrable couplings and tri integrable couplings

the deduction of the lax representation for constrained flows from the adjoint representation for x and tn finite dimensional hamiltonian systems obtained from the decompositions of zero curvature equations it is shown that their lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems as a consequence the zero curvature representation for soliton hierarchy with source is presented

the strict positivstellensatz for global analytic functions and the moment problem for semianalytic sets analytic functions strictly positive on a global semianalytic set x f1 0 fk 0 in rn are characterized as functions expressible as g a0 a1f1 akfk for strictly positive global analytic functions a0 ak the proof is elementary using the fact that the analytic functions are dense in c rn r in the whitney topology the same proof works for nash functions this is an improvement of the standard analytic version of stengle s positivstellensatz in two directions the hypothesis is weaker there is no requirement that x be compact and the conclusion is stronger several applications are given including i a new proof of the weak positivstellensatz for semianalytic sets and ii the solution of thek moment problem for basic closed semianalytic

maximal entropy odd orbit types a periodic orbit of a continuous map of an interval induces in a natural way a cyclic permutation called its type we consider a family of orbit types of period n congruent to 1 mod 4 introduced recently by misiurewicz and nitecki we prove that the misiurewicz nitecki orbit types and their natural generalizations to the remaining odd periods n have maximal entropy among all orbit types of period n and even among all n permutations

canonical structure of soliton equations i abstract the canonical structure of the nonlinear evolution equations in 1 1 dimensions solvable in terms of an n n inverse scattering problem is discussed the simplest form of the scattering problems that is those containing the spectral parameter linearly is considered it applies to most of the known soliton equations like the korteweg de vries eq the sine gordon eq and the boussinesq eq discussion of various possible reductions of the number of dependent variables by imposing constraints consistent with the hamiltonian flows is given together with the canonical structure of the reduced systems a direct proof of the involutive character of the infinite number of conserved quantities is given for the general case as well as the reduced case the relation between the conserved quantities and symmetry transformations lie backlund transformations becomes very simple in this framework

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

an integrable extension of a soliton hierarchy associated with so 3 ℝ and its bi hamiltonian formulation a hierarchy of soliton equations is constructed from an extended spectral problem associated with the three dimensional special orthogonal real lie algebra so 3 by means of zero curvature equations the bi hamiltonian structure of this hierarchy is also presented via the trace identity

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

a soliton hierarchy associated with so 3 r we generate a hierarchy of soliton equations from zero curvature equations associated with the real lie algebra so 3 r and show that each equation in the resulting hierarchy has a bi hamiltonian structure and thus integrable in the liouville sense

closure of the cone of sums of 2d powers in commutative real topological algebras abstract let r be a unitary commutative r algebra and k x r hom r r closed with respect to the product topology we consider r endowed with the topology t k induced by the family of seminorms a a for k and a r in case k is compact we also consider the topology induced by a k sup k a for a r if k is zariski dense then those topologies are hausdorff in this paper we prove that the closure of the cone of sums of 2d powers r 2 d with respect to those two topologies is equal to psd k a r a 0 for all k in particular any continuous linear functional l on the polynomial ring r r x r x 1 x n with l h 2 d 0 for each h r x is integration with respect to a positive borel measure supported on k finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies

an integrable coupling hierarchy of the mkdv_integrable systems its hamiltonian structure and corresponding nonisospectral integrable hierarchy a four by four matrix spectral problem is introduced locality of solution of the related stationary zero curvature equation is proved an integrable coupling hierarchy of the mkdv integrable systems is presented the hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity it is shown that the resulting integrable couplings are all liouville integrable hamiltonian systems ultimately through the nonisospectral zero curvature representation a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

symmetry broken canonizations of the semi discrete integrable nonlinear schrodinger system with background controlled inter site coupling the integrable system of nonlinear schrodinger type consisting of two mutually symmetric basic subsystems and one concomitant subsystem given on a ribbon of triangular lattice is characterized by the highly nonstandard form of poisson structure the fact of system criticality against the positive semi definite background parameter provides us the clue how to organize the nonlinear point transformation ensuring the canonization of field variables precisely it allows to introduce two subsets of mutually symmetric intermediate auxiliary field variables and to involve one or another of them into a rational procedure of canonization realizable already in terms of two mutually asymmetric subsets of field amplitudes there are two variants of system asymmetric standardization namely the minus asymmetric and plus asymmetric ones emanated from the two leg structure of underlying space lattice of the primary noncanonical nonlinear system either of the two asymmetrically standardized nonlinear systems consi

a hierarchy of nonlinear evolution equations and finite dimensional involutive systems a spectral problem and an associated hierarchy of nonlinear evolution equations are presented in this article in particular the reductions of the two representative equations in this hierarchy are given one is the nonlinear evolution equation rt rx 2i r2 r which looks like the nonlinear schrodinger equation the other is the generalized derivative nonlinear schrodinger equation rt 1 2i rxx i r 2r r 2r x r 2rx 2i 2 r 4r which is just a combination of the nonlinear schrodinger equation and two different derivative nonlinear schrodinger equations d j kaup and a c newell j math phys 19 789 1978 m j ablowitz a ramani and h segur j math phys 21 1006 1980 the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system under a constraint between the potentials and the spectral functions at the end of this article the involutive solutions of the hierarchy of nonlinear evolution equations are obtained particularly the involutiv

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

lax representation r matrix method and separation of variables for the neumann type restricted flow the r matrix method and the method of separation of variables for restricted flows with neumann constraints are considered restricted jaulent miodek flows are taken as illustrative examples lax representations are obtained from the adjoint representation of the corresponding soliton hierarchy the r matrix formula based on this lax representation is derived and the separation of variables for the hamilton jacobi equation are performed

stratification of continuous maps of an interval we define the notion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points we define also strongly simple orbits and show in particular that they represent periodic orbits with minimum entropy further results are obtained for unimodal maps with negative schwarzian which sharpen recent results of block and hart

zero curvature representation bi hamiltonian structure and an integrable hierarchy for the zakharov ito system in the present paper we present an integrable hierarchy for the zakharov ito system we first construct the lenard recursion sequence and zero curvature representation for the zakharov ito system following cao s method as significantly generalized by other authors we then construct the bi hamiltonian structures employing variational trace identities but woven together with the lenard recursion sequences from this we are in a position to construct an integrable hierarchy of equations from the zakharov ito system and we obtain the recursion operator and poisson brackets for constructing this hierarchy finally we demonstrate that the obtained hierarchy is indeed liouville integrable

primary cycles on the circle we consider cycles i e periodic orbits of continuous degree one maps of the circle one cycleaforces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second we characterize primary cycles i e those which force no other cycle of the same period

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

an integrable generalization of the d kaup newell soliton hierarchy and its bi hamiltonian reduced hierarchy abstract we present a new spectral problem a generalization of the d kaup newell spectral problem associated with the lie algebra sl 2 r zero curvature equations furnish the soliton hierarchy the trace identity produces the hamiltonian structure for the hierarchy and shows its liouville integrability lastly a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi hamiltonian structure the major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high order symmetries

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

two soliton hierarchies associated with so 4 and the applications of su 2 su 2 so 4 based on the six dimensional real special orthogonal lie algebra so 4 we construct two different isospectral problems and the two hierarchies of soliton equations from zero curvature equations can be generated at last we use the property of the isomorphism su 2 su 2 so 4 and obtain the relations between the hierarchy of su 2 su 2 and the hierarchy of so 4

binary symmetry constraints of n wave interaction equations in 1 1 and 2 1 dimensions binary symmetry constraints of the n wave interaction equations in 1 1 and 2 1 dimensions are proposed to reduce the n wave interaction equations into finite dimensional liouville integrable systems a new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix lax operator and used to show the liouville integrability of the constrained flows of the n wave interaction equations the constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the n wave interaction equations and thus the integrability by quadratures are shown for the n wave interaction equations by the constrained flows

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

a generating scheme for conservation laws of discrete zero curvature equations and its application abstract a generating scheme for conservation laws of discrete integrable equations arising as discrete zero curvature equations is presented from pairs of associated matrix spectral problems an illustrative application to the volterra lattice equation is made through a pair of 2 2 matrix spectral problems the general theory is not limited to 2 2 matrix spectral problems but for arbitrary order matrix spectral problems

a family of completely integrable multi hamiltonian systems explicitly related to some celebrated equations by introducing a spectral problem with an arbitrary parameter we derive a kaup newell type hierarchy of nonlinear evolution equations which is explicitly related to many important equations such as the kundu equation the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the burgers equation the modified korteweg devries mkdv equation and the sharma tasso olver equation it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure under the bargann constraint between the potentials and the eigenfunctions the spectral problem is nonlinearized as a finite dimensional completely integrable hamiltonian system the involutive representation of the solutions for the kaup newell type hierarchy is also presented in addition an n fold darboux transformation of the kundu equation is constructed with the help of its lax pairs and a reduction technique according to the darboux transformation the solutions of the ku

generalization of sturm liouville theory to a system of ordinary differential equations with dirac type spectrum the sturm liouville theory is generalized to dirac equation like systems of ordinary differential equations it is shown how the comparison theorem and conversion to integral equations can be generalized

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

trigonometric polynomials positive on frequency domains and applications to 2 d fir filter design we propose a characterization of multivariate trigonometric polynomials that are positive on a given frequency domain the positive polynomials are parameterized as a linear function of sum of squares polynomials and so semidefinite programming sdp is applicable the frequency domain is expressed via the positivity of some trigonometric polynomials we also give a bounded real lemma brl in which a bounding condition on the magnitude of the frequency response of a multidimensional finite impulse response fir filter is expressed as a linear matrix inequality lmi this brl avoids the problem of a lack of spectral factorization in the multidimensional case all the proposed theoretical contributions can be implemented only as sufficient conditions due to degree limitations on the sum of square polynomials however the two dimensional 2 d fir filter designs we study numerically suggest that these limitations have negligible impact on the optimality

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

an approach for constructing nonisospectral hierarchies of evolution equations for a given isospectral lambda t 0 hierarchy of evolution equations the author proposes a simple method of constructing its corresponding nonisospectral lambda t lambda n n or 0 hierarchy of evolution equations closely related to tau symmetries it is crucial to find an initial lax operator w0 and an initial vector field g0 satisfying the key equation w0 l l g0 i in which l i are spectral and identity operators respectively as examples the author presents the corresponding nonisospectral hierarchies of equations and displays the fundamental relations generating symmetry algebras for the kdv hierarchy the akns hierarchy and a new integrable hierarchy

a coupled akns kaup newell soliton hierarchy a coupled akns kaup newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from hamiltonian pairs zero curvature representations and tri hamiltonian structures are established for all coupled akns kaup newell systems in the hierarchy therefore all systems have infinitely many commuting symmetries and conservation laws two reductions of the systems lead to the akns hierarchy and the kaup newell hierarchy and thus those two soliton hierarchies also possess tri hamiltonian structures

dirac reduction for poisson vertex algebras we construct an analogue of dirac s reduction for an arbitrary local or non local poisson bracket in the general setup of non local poisson vertex algebras this leads to dirac s reduction of an arbitrary non local poisson structure we apply this construction to an example of a generalized drinfeld sokolov hierarchy