new liouville integrable noncanonical hamiltonian systems from the akns spectral problem liouville integrable noncanonical hamiltonian systems with variable coefficient symplectic forms are generated from the akns 3 3 matrix lax pairs by binary symmetry constraints of the akns hierarchy these integrable systems provide new integrable factorization for every akns system in the hierarchy

a borcherds kac moody superalgebra with conway symmetry we construct a borcherds kac moody bkm superalgebra on which the conway group hbox co 0 acts faithfully we show that the bkm algebra is generated by the physical states brst cohomology classes in a chiral superstring theory we use this construction to produce denominator identities for the chiral partition functions of the conway module v s natural a supersymmetric c 12 chiral conformal field theory whose twisted partition functions enjoy moonshine properties and which has automorphism group isomorphic to hbox co 0 in particular these functions satisfy a genus zero property analogous to that of monstrous moonshine finally we suggest how one may promote the denominators to spacetime bps indices in type ii string theory which might thus furnish a physical explanation of the genus zero property of conway moonshine

the liouville integrable systems associated with a new discrete 3 3 matrix spectral problem a new discrete 3 times 3 matrix spectral problem with three potentials is introduced and the corresponding family of liouville integrable lattice equations is obtained by applying the discrete trace identity it is shown that the hierarchy possesses a hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals finally infinitely many conservation laws of corresponding lattice systems are obtained by a direct way

multiple results of p laplacian dynamic equations on time scales in this paper a new fixed point theorem introduced by avery and peterson and another fixed point theorem are applied to boundary value problems for p laplacian dynamic equations on time scales by imposing some growth conditions on the nonlinearity f we establish the existence results of at least three positive solutions as an application some examples are included to illustrate the main results

continuous limits for an integrable coupling of the kac van moerbeke hierarchy in this paper we present an integrable coupling of lattice hierarchy and its continuous limits by using lie algebra sl 4 by introducing a complex discrete spectral problem the integrable coupling of kac van moerbeke lattice hierarchy is derived it is shown that a new complex lattice spectral problem converges to the integrable coupling of discrete soliton equation hierarchy which has the integrable coupling system of mkdv hierarchy as a new kind of continuous limit

integrable coupling of the ablowitz ladik hierarchy and its hamiltonian structure abstract the trace identity is generalized to work for the discrete zero curvature equation associated with the lie algebra possessing degenerate killing forms then a kind of integrable coupling of the ablowitz ladik al hierarchy is obtained and its hamiltonian structure is worked out moreover liouville integrability of the integrable coupling is demonstrated

eigenvalue problems for nonlinear differential equations on a measure chain abstract values of are determined for which there exist positive solutions of the second order differential equation on a measure chain u t a t f u t 0 t 0 1 satisfying either the conjugate boundary conditions u 0 u 1 0 or the right focal boundary conditions u 0 u 1 0 where a and f are positive valued and both lim x 0 f x x and lim x f x x exist

existence of solutions for third order boundary value problems on a time scale abstract let t be a closed subset of r with inf t and sup t for the third order nonlinear equation u 3 f t u u u t t where f t r3 r is continous we assume solutions of initial value problems are unique and extend to t we consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on t

a bi hamiltonian formulation for triangular systems by perturbations a bi hamiltonian formulation is proposed for triangular systems resulting from perturbations around solutions from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed provided that one operator of the hamiltonian pair is invertible through our formulation four examples of triangular systems are exhibited which also show that bi hamiltonian systems in both lower dimensions and higher dimensions are many and varied two of four examples give local 2 1 dimensional bi hamiltonian systems and illustrate that multiscale perturbations can lead to higher dimensional bi hamiltonian systems

integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of lie algebras abstract staring from a new spectral problem a hierarchy of the soliton equations is derived it is shown that the associated hierarchies are infinite dimensional integrable hamiltonian systems by the procedure of nonlinearization of the lax pairs the integrable decomposition of the whole soliton hierarchy is given further we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of lie algebras

hamiltonian and super hamiltonian systems of a hierarchy of soliton equations abstract by considering an isospectral eigenvalue problem a hierarchy of soliton equations are derived two types of extensions are presented by enlarging the associated spectral problem with the aid of generalized trace identity and the super trace identity the hamiltonian and super hamiltonian structures for the integrable extensions are established

second order periodic boundary value problems on time scales in this paper we study the existence of solutions of periodic boundary value problem for second order nonlinear dynamic equations on time scales for this purpose we use two methods here first sign properties of the green s function are investigated and existence results of bounded solutions of nonlinear boundary value problems are established as a result of schauder fixed point theorem second the monotone method is discussed to assure the existence and uniqueness of solutions of such periodic boundary value problems on time scales

an integrable system and associated integrable models as well as hamiltonian structures starting from an existed lie algebra introduces a new lie algebra a1 e1 e2 e3 so that two isospectral lax matrices are established by employing the tu scheme an integrable equation hierarchy denoted by ieh is obtained from which a few reduced evolution equations are presented one of them is the mkdv equation the elliptic variable solutions and three kinds of darboux transformations for one coupled equation which is from the ieh are worked out respectively finally we take use of the lie algebra a1 to generate eight higher dimensional lie algebras from which the linear integrable couplings the nonlinear integrable couplings and the bi integrable couplings of the ieh are engendered whose hamiltonian structures are also obtained by the variational identity then further reduce one coupled integrable equation to get a nonlinear generalized mkdv equation

an integrable couplings of g wki equations hierarchy with self consistent sources we present a kind of integrable couplings of soliton equations hierarchy with self consistent sources by using the loop algebra sl 4 as an application a new integrable couplings of the generalized wadati konono ichikawa g wki equations hierarchy with self consistent sources is derived through the enlarged loop algebra sl 4

new super integrable hierarchies associated with osp 2 2 and spo 2 2 and their applications abstract based on the lie superalgebras osp 2 2 and spo 2 2 we generate some new super soliton hierarchies these hierarchies can reduce to the akns hierarchy and the super akns hierarchy and by using the supertrace identity on lie superalgebras we construct the super hamiltonian structures of these new super soliton hierarchies then as the applications of spo 2 2 osp 2 2 and spo 2 2 sl 1 2 we obtain the relations of different hierarchies associated with different lie superalgebras

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 super integrable hamiltonian couplings based on a kind of new lie superalgebras we proposed a scheme for constructing nonlinear super integrable couplings super trace identity over the corresponding loop superalgebras is used to furnish the super hamiltonian structures for the resulting nonlinear super integrable couplings we illustrate the application of the scheme by means of an enlarged super ablowitz kaup newell segur akns matrix spectral problem and present a nonlinear super integrable hamiltonian couplings for the super akns hierarchy

a new loop algebra and a corresponding integrable hierarchy as well as its integrable coupling a type of new interesting loop algebra gm m 1 2 with a simple commutation operation just like that in the loop algebra a1 is constructed with the help of the loop algebra gm a new multicomponent integrable system m akns kn hierarchy is worked out as reduction cases the m akns hierarchy and m kn hierarchy are engendered respectively in addition the system 1 akns kn which is a reduced case of the m akns kn hierarchy above is a unified expressing integrable model of the akns hierarchy and the kn hierarchy obviously the m akns kn hierarchy is again a united expressing integrable model of the multicomponent akns hierarchy m akns and the multicomponent kn hierarchy m kn this article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations finally we work out an integrable coupling of the m akns kn hierarchy

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

an integrable coupling family of the toda lattice systems its bi hamiltonian structure and a related nonisospectral integrable lattice family starting from a four by four matrix spectral problem an integrable coupling family of the toda lattice systems is derived a pair of discrete hamiltonian operators is presented and a sequence of the corresponding hamiltonian functionals is constructed by using the discrete variational identity then the bi hamiltonian structure of the obtained integrable coupling family is established finally a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

stochastic machine minimization with constant service times we consider the nonpreemptive scheduling of n 1 jobs on identical parallel machines with job running times all given by some constant the objective is to minimize the expected number of machines needed throughout a schedule subject to the following waiting time constraints at time 0 a timer with random initial value w is started after time w all jobs must either be finished or running on a machine the value of w is not known in advance but its distribution is made available to the scheduler in general if job running times are random there might be situations in which after running jobs on a given number of machines it would be desirable to activate new machines because the remaining times of unfinished jobs are stochastically too large compared to the time remaining on the timer a principal result of this paper is that randomness of job running times is essential to such situations if running times are deterministic then irrespective of the distribution of w an optimal policy initially assigns jobs to some number of machines kn which depends on the distribution of w and maintains that number while there are jobs available until the timer has expired at which point any remaining as yet unscheduled jobs are assigned to new unused machines this paper also analyzes the dependence of kn and the minimal expected cost on the parameters of the timer distribution when w is either exponentially distributed or uniformly distributed on a finite interval the determination of kn turns out to have an intriguing number theoretic flavor

the quadratic form identity for constructing the hamiltonian structures of the discrete integrable systems the quadratic form identity is extended to the discrete version which can be used to construct the hamiltonian structures of the discrete integrable systems associated with the lie algebra possessing degenerate killing forms especially it can be used to work out the hamiltonian structures of some kinds of discrete integrable couplings then a kind of integrable coupling of the toda hierarchy is obtained and its hamiltonian structure is worked out by using the discrete quadratic form identity moreover the liouville integrability of the integrable coupling is demonstrated

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem with all derivatives in this paper firstly some errors in the proof of our paper several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales appl math comput 190 2007 566 575 are pointed and we make the corresponding correction when t r then the more general problem with all derivatives is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence and uniqueness of nontrivial solution are obtained by using leray schauder nonlinear alternative and banach fixed point theorem

integrable semi discretization of the coupled modified kdv equations the discrete version of the inverse scattering method proposed by ablowitz and ladik is extended to solve multi component systems the extension enables one to solve the initial value problem which proves directly the complete integrability of a semi discrete version of the coupled modified korteweg de vries kdv equations and their hierarchy it also provides a procedure to obtain conservation laws and multi soliton solutions of the hierarchy

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

two integrable couplings of the tu hierarchy and their hamiltonian structures the double integrable couplings of the tu hierarchy are worked out by use of vector loop algebras g 6 and g 9 respectively also the hamiltonian structures of the obtained system are given by the quadratic form identity

constructing integrable coupling systems of hamiltonian lattice equations using semi direct sums of lie algebras abstract in this article by considering a discrete isospectral problem a hierarchy of hamiltonian lattice equations are derived two types of semi direct sums of lie algebras are proposed using which a practicable way to construct discrete integrable couplings is introduced as an application two kinds of discrete integrable couplings of the resulting system are worked out

an integrable lattice hierarchy associated integrable coupling darboux transformation and conservation laws abstract a new integrable lattice hierarchy is constructed from a discrete matrix spectral problem some related properties of the new hierarchy are discussed the hamiltonian structures and liouville integrability of the new hierarchy are established by using the discrete trace identity a kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems a darboux transformation dt with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its lax representation the explicit solutions are obtained via the dt the structures for those solutions are graphically investigated all these properties might be helpful to understanding some physical phenomena

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

two integrable couplings of a generalized d kaup newell hierarchy and their hamiltonian and bi hamiltonian structures abstract we enlarge the spectral problem of a generalized d kaup newell d kn spectral problem solving the enlarged zero curvature equations we produce integrable couplings a reduction of the spectral matrix leads to a second integrable coupling system next bilinear forms that are symmetric ad invariant and non degenerate on the given non semisimple matrix lie algebra are computed to employ the variational identity the variational identity is then applied to the original enlarged spectral problem of a generalized d kn hierarchy and the reduced problem hamiltonian structures are presented as well as a bi hamiltonian formulation of the reduced problem both hierarchies have infinitely many commuting high order symmetries and conserved densities i e are liouville integrable

topics in quantum integrable systems recent developments in the theory of quantum integrable particle systems in one dimension with inverse square interactions are reviewed first the yangian symmetry is introduced and the energy spectra of the related spin models are discussed the character of the su n 1 wznw theory is shown to be closely related with the rogers szego polynomial second the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation is given based on the representation the dunkl operators associated with the classical root systems are constructed the macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case finally some results on short range interacting systems are mentioned

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

a family of liouville integrable lattice equations and its conservation laws abstract a new discrete spectral problem of matrix three by three with three potentials is introduced and by using the discrete trace identity a hierarchy of liouville lattice equations is obtained the infinite number of conservation laws of the hierarchy of hamiltonian system are presented

integrable couplings of vector akns soliton equations by enlarging the associated matrix spectral problems two specific classes of multicomponent integrable couplings of the physically important vector akns soliton equations are constructed which can be linked to each other by a backlund transformation the resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures and thus each system in the two hierarchies has infinitely many commuting symmetries

four lie algebras associated with r6 and their applications the first part in the paper reads that a three dimensional lie algebra is first introduced whose corresponding loop algebra is constructed for which isospectral problems are established by employing zero curvature equations a modified kaup newell mkn soliton hierarchy of evolution equations is obtained the corresponding hereditary operator and hamiltonian structure are worked out respectively then two types of enlarging semisimple lie algebras isomorphic to the linear space r6 are followed to construct one of them is a complex lie algebra their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mkn hierarchy the hereditary operators hamiltonian structures of the hierarchies are produced again respectively the exact computing formulas of the constant appearing in the trace identity and the variational identity are derived under the semisimple algebras the second part of this pap

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

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

existence of a solution and a positive solution of a boundary value problem for a nonlinear fourth order dynamic equation abstract in this paper by using the schauder fixed point theorem we offer some existence criteria for a solution and a positive solution to the following fourth order two point boundary value problem on time scale t u t f t u t u t 0 t a 2 b u a a u 2 b b u a c u b d where a b t and a b

the generalized kupershmidt deformation for the fifth order coupled kdv equations hierarchy abstract based on the kupershmidt deformation we propose the generalized kupershmidt deformation gkd to construct new systems from integrable bi hamiltonian system as applications the generalized kupershmidt deformation of the fifth order coupled kdv equations hierarchy with self consistent sources and its lax representation are presented

the multicomponent generalized kaup newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions we devise a new simple loop algebra gm and an isospectral problem by making use of the tu scheme the multicomponent generalized kaup newell hierarchy is obtained furthermore an expanding loop algebra fm of the loop algebra gm is presented based on fm the multicomponent integrable couplings system with two arbitrary functions of the multicomponent generalized kaup newell hierarchy are worked out the method can be applied to other nonlinear evolution equations hierarchy

existence of solution and positive solution of bvp for nonlinear third order dynamic equation abstract in this paper we are concerned with the following third order two point boundary value problem on time scale t u t f t u t u t 0 t a b u a a u 2 b b u a c some existence criteria of solution and positive solution are established by using leray schauder fixed point theorem and our main conditions are local an example is also included to illustrate the importance of the results obtained

existence of solution for discontinuous third order boundary value problems abstract in this paper we obtain existence results for the problem u q u f t u with boundary conditions u a a u b b u a c and u a u b u a u b u a c we assume f a caratheodory function q l r 0 such that 1 q l loc r 0 and suppose the existence of lower and upper solutions the existence of solution for the first considered conditions is obtained as a consequence of the fixed points theorems we obtain the solution of the second problem as a limit of solutions of the first case for the first problem the monotone method is developed

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

a non isospectral integrable couplings of volterra lattice hierarchy with self consistent sources a kind of non isospectral integrable couplings of discrete soliton equations hierarchy with self consistent sources associated with email protected 4 y f zhang e g fan characteristic numbers of matrix lie algebras commun theor phys china 49 2008 845 is presented as an application example the hierarchy of non isospectral cubic volterra lattice hierarchy with self consistent sources is derived furthermore we construct a non isospectral integrable couplings of cubic volterra lattice hierarchy with self consistent sources by using the loop algebra email protected 4

an integrable couplings of dirac soliton hierarchy with self consistent sources abstract a kind of the integrable couplings of soliton equations hierarchy with self consistent sources associated with sl 4 is presented as an application example we construct a new integrable couplings of the dirac soliton hierarchy with self consistent sources by using of loop algebra sl 4

the processor minimization problem with independent waiting time constraints abstract this paper studies a dual of classical stochastic scheduling of parallel processor systems the given data are n jobs with running times t 1 t n and waiting time constraints w 1 w n these sequences are independent and each consists of i i d random variables although the t i and w i are not known in advance they are known to be samples from exponential distributions with given parameters scheduling policies are nonpreemptive and have the option for each i of making the i th job wait during 0 w i however by the random deadline w i the i th job must be assigned to a processor if it is still waiting in this paper we find a policy that minimizes the expected number of processors used among all policies of the above type we also estimate the expected number of processors used under this policy there appears to be no simple exact formula for this quantity so we turn to an asymptotic analysis based on a continuous approximation

solutions for higher order dynamic equations on time scales let t be a time scale we study the existence of positive solutions for the nonlinear four point singular boundary value problem with higher order p laplacian dynamic equations on time scales by using the fixed point index theory the existence of positive solution and many positive solutions for nonlinear four point singular boundary value problem with p laplacian operator are obtained

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

new factorization of the kaup newell hierarchy abstract the new factorizations of each equation in the kaup newell kn hierarchy into two commuting x and t n finite dimensional integrable hamiltonian systems fdihs via the higher order potential eigenfunction constraints are presented the integrability commutativity and lax representation for these fdihss are deduced from the adjoint representations of the auxiliary linear problems for the kn hierarchy this approach provides a natural way to study soliton hierarchies with source the zero curvature representation and darboux transformation for the kn hierarchy with source related to eigenfunctions are found

nonlinearization of spectral problems for the perturbation kdv systems abstract the theory of nonlinearization of spectral problems is developed for investigating the perturbation kdv systems and a kind of finite dimensional integrable hamiltonian systems is generated from spectral problems of the perturbation kdv systems to establish the liouville integrability of the obtained hamiltonian systems we identify them with the perturbation systems of the garnier system which can be directly proved to be integrable a useful formula to compute the functional gradient of the spectral parameter with respect to the potential is also presented for arbitrary order matrix spectral problems

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

analytic foliations and the theory of levels let m be a closed n manifold a transversely orientable real analytic foliation of m by leaves of dimension n 1 in this note we establish a basic structure theorem 5 1 for such foliations obtaining as a corollary 5 2 a result about growth types in real analytic foliations that was announced in c c2 6 13 this growth result has also been announced by tsuchiya its 5 1 and is known independently by hector hesitancy about publishing this material at least on the part of the present authors was due to a fear that the general structure theorem 5 1 might be practically empty indeed that theorem describes the architecture of transversely analytic foliations via the theory of levels c c1 and no examples were known of such foliations with leaves at levels greater than 1 now however by methods of hirsch hi it is quite easy to show that most of the structure allowed by 5 t does in fact occur we describe suitable constructions in sect 7 we assume familiarity with the theory of levels in c z foliations i c c1 c c2 in particular we recall the following

optimal stochastic allocation of machines under waiting time constraints the nonpreemptive scheduling of n geqslant 1 stochastic jobs is considered to minimize the expected number of parallel machines needed to meet given waiting time constraints the number of machines available is unlimited the running times of the jobs are denoted t 1 ldots t n and are taken to be independent samples of an exponentially distributed random variable t with mean 1 job waiting times are to be bounded stochastically by a nonnegative random variable w independent of t 1 ldots t n at time zero a timer is started with an initial value w and job scheduling begins when the timer expires all jobs still waiting for a machine are assigned to available machines only the distributions of w and the job durations t 1 ldots t n are known to the scheduler in advance a scheduling policy is defined which is proven to be optimal when w has an exponential distribution and is asymptotically optimal as n to infty when w is a constant hard deadline in the exponential case

positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem the two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of lax operators with respect to the spectral parameter respectively moreover the integrable lattice models in the positive hierarchy are of polynomial type and the integrable lattice models in the negative hierarchy are of rational type further we construct three integrable coupling systems of the positive hierarchy through enlarging lax pair method

a nonlinear integrable couplings of c kdv soliton hierarchy and its infinite conservation laws abstract a kind of nonlinear integrable couplings of c kdv soliton hierarchy is constructed then we present the infinitely many conservation laws for the nonlinear integrable couplings of the c kdv soliton hierarchy

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

hierarchies integrable decompositions and conservation laws of geng equation abstract in this paper two hierarchies of the geng equations are presented including positive non isospectral hierarchy and negative non isospectral hierarchy moreover integrable couplings of the corresponding geng hierarchies are also constructed by enlarging the associated matrix spectral problem three new integrable decompositions and conservation laws of the isospectral geng equation are also obtained the gauge transformations are used to obtain the associated binary symmetry constraints of the geng equation at the first time

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

a 3 sx 3 matrix spectral problem for akns hierarchy and its binary nonlinearization abstract a 3 3 matrix spectral problem for akns soliton hierarchy is introduced and the corresponding bargmann symmetry constraint involving lax pairs and adjoint lax pairs is discussed an explicit new poisson algebra is proposed and thus the liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system the obtained nonlinearized lax systems in which the nonlinearized spatial system is intimately related to stationary akns flows lead to a sort of new involutive solutions to each akns soliton equation therefore the binary nonlinearization theory is successfully extended to a case of 3 3 matrix spectral problem for akns hierarchy

on a conjecture by coffman flatto and wright on stochastic machine minimization we investigate a conjecture stated by coffman flatto and wright within the context of a stochastic machine minimization problem with a hard deadline we prove that the conjecture is true

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

triple positive solutions of m point bvps for p laplacian dynamic equations on time scales abstract this paper is concerned with the existence of positive solutions of p laplacian dynamic equation p u t a 1 t f u t 0 subject to boundary conditions u 0 b 0 i 1 m 2 a i u i 0 u t 0 or u 0 0 u t b 1 i 1 m 2 b i u i 0 where p v v p 2 v with p 1 by using the five functionals fixed point theorem we prove that the boundary value problem has at least three positive solutions as an application an example is given to illustrate the result

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

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales abstract in this paper a nonlinear higher order three point boundary value problem on time scales is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence of nontrivial solution are obtained by using leray schauder nonlinear alternative as an application some examples to demonstrate our results are given