hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

unconventional schemes for a class of ordinary differential equations with applications to the korteweg de vries equation an unconventional numerical method for solving a restrictive and yet often encountered class of ordinary differential equations is proposed the method has a crucial what we callreflexive property and requires solving one linear system per time step but is second order accurate a systematical and easily implementable scheme is proposed to enhance the computational efficiency of such methods whenever needed applications are reported on how the idea can be applied to solve the korteweg de vries equation discretized in space

second variation and stabilities of minimal lagrangian submanifolds in kahler manifolds first we derive a new second variation formula which holds for any minimal lagrangian submanifold in k ihler manifolds using this we prove that any minimal lagrangian submanifold in kfihler manifolds with nonpositive chern class is stable we also prove that in kfihler manifolds with positive chern class any stable minimal lagrangian submanifold l satisfies that h l 0 next we introduce the notion of hamiltonian stability either local or global of minimal lagrangian submanifolds in einstein kfihler manifolds using our second variation formula we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in einstein k ihler manifolds this criterion is satisfied for all real forms which are einstein of irreducible hermitian symmetric spaces e g the cannonical rp c cp it is also satisfied by the clifford torus t c cp

dft calculation of the generalized and drazin inverse of a polynomial matrix a new algorithm is presented for the determination of the a generalized inverse and b drazin inverse of a polynomial matrix the proposed algorithms are based on the discrete fourier transform and thus are computationally fast in contrast to other known algorithms the above algorithms are implemented in the mathematica programming language and illustrated via examples

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

computation of the generalized inverse of a polynomial matrix and applications abstract the computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its laurent expansion of a nonregular polynomial matrix the proposed algorithm constitutes a generalization of the algorithm proposed by fragulis et al for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis

integrable reductions of the bogoyavlenskij itoh lotka volterra systems given a constant skew symmetric matrix a it is a difficult open problem whether the associated lotka volterra system is integrable or not we solve this problem in a special case when a is a toeplitz matrix where all off diagonal entries are plus or minus one in this case the associated lotka volterra system turns out to be a reduction of liouville integrable systems whose integrability was shown by bogoyavlenskij and itoh we prove that the reduced systems are also liouville integrable and that they are also non commutative integrable by constructing a set of independent first integrals having the required involutive properties with respect to the poisson bracket these first integrals fall into two categories one set consists of polynomial functions that are restriction of the bogoyavlenskij itoh integrals their involutivity was already pointed out by bogoyavlenskij the other set consists of rational functions which are obtained through a poisson map from the first integrals of some recently d

involutivity of integrals of sine gordon modified kdv and potential kdv maps closed form expressions in terms of multi sums of products have been given in tran et al 2009 j phys a math theor 42 225201 and van der kamp et al 2007 j phys a math theor 39 12789 98 of integrals of sine gordon modified korteweg de vries and potential korteweg de vries maps obtained as so called p 1 travelling wave reductions of the corresponding partial difference equations we prove the involutivity of these integrals with respect to recently found symplectic structures for those maps the proof is based on explicit formulae for the poisson brackets between multi sums of products

geometric properties of kahan s method we show that kahan s discretization of quadratic vector fields is equivalent to a runge kutta method when the vector field is hamiltonian on either a symplectic vector space or a poisson vector space with constant poisson structure the map determined by this discretization has a conserved modified hamiltonian and an invariant measure a combination previously unknown amongst runge kutta methods applied to nonlinear vector fields this produces large classes of integrable rational mappings in two and three dimensions explaining some of the integrable cases that were previously known

liouville integrability and superintegrability of a generalized lotka volterra system and its kahan discretization we prove the liouville and superintegrability of a generalized lotka volterra system and its kahan discretization

a new method for computing the solutions of differential equation systems using generalized inverse via maple in j jones et al j symbolic comput 25 1998 99 124 an algorithm for computing the generalized inverse of a singular rational matrix has been implemented in maple in this paper we present a new symbolic algorithm to compute the initial value problem of higher order differential equations with constant coefficients using generalized inverse the algorithm is subsequently implemented in the symbolic computational package maple

on the existence of hamiltonian stationary lagrangian submanifolds in symplectic manifolds let m omega be a compact symplectic 2n manifold and g a riemannian metric on m compatible with omega for instance g could be k ahler with k ahler form omega consider compact lagrangian submanifolds l of m we call l it hamiltonian stationary or it h minimal if it is a critical point of the volume functional mathop rm vol nolimits g under hamiltonian deformations computing mathop rm vol nolimits g l using g vert l it is called it hamiltonian stable if in addition the second variation of mathop rm vol nolimits g under hamiltonian deformations is nonnegative our main result is that if l is a compact hamiltonian stationary lagrangian in mathbin bbb c n which is it hamiltonian rigid then for any m omega g as above there exist compact hamiltonian stationary lagrangians l in m contained in a small ball about some p in m and locally modelled on t l for small t0 identifying m near p with mathbin bbb c n near 0 if l is hamiltonian stable we can take l to be hamiltonian stable applying this to known examples l in mathbin bbb c n shows that there exist families of hamiltonian stable hamiltonian stationary lagrangians diffeomorphic to t n and to cal s 1 times cal s n 1 mathbin bbb z 2 and with other topologies in every compact symplectic 2n manifold m omega with compatible metric g

hyperelliptic prym varieties and integrable systems we introduce two algebraic completely integrable analogues of the mumford systems which we call hyperelliptic prym systems because every hyperelliptic prym variety appears as a fiber of their momentum map as an application we show that the general fiber of the momentum map of the periodic volterra lattice is an affine part of a hyperelliptic prym variety obtained by removing n translates of the theta divisor and we conclude that this integrable system is algebraic completely integrable

computation of the inverse of a polynomial matrix and evaluation of its laurent expansion abstract an algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil se a due to mertzios 1984 in the case of general systems described in differential operator form recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix as well as for the coefficients of the characteristic polynomial of the polynomial matrix a simple method is also presented that allows the evaluation of the laurent expansion for the inverse of a polynomial matrix of any order specifically an arma model is given for the computation of the coefficient matrices of the laurent expansion in terms of the coefficient matrices of the polynomial matrix without inverting the latter the laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular or generalized systems

on the regularity of hamiltonian stationary lagrangian submanifolds abstract we prove a morrey type theorem for hamiltonian stationary lagrangian submanifolds of c n if a c 1 lagrangian submanifold is a critical point of the volume functional under hamiltonian variations then it must be real analytic locally a hamiltonian stationary lagrangian submanifold is determined geometrically by harmonicity of its lagrangian phase function or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the hamiltonian stationary lagrangian submanifolds locally our result shows that morrey s theorem for minimal submanifolds admits a complete fourth order analogue we establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with c 1 1 norm below a dimensional constant and to c 1 1 potential functions under certain convexity conditions whose lagrangian phase functions are weakly harmonic

integrability properties of kahanʼs method we present several novel examples of integrable quadratic vector fields for which kahan s discretization method preserves integrability our examples include generalized suslov and ishii systems nambu systems riccati systems and the first painleve equation we also discuss how manin transformations arise in kahan discretizations of certain vector fields

integrable and superintegrable systems associated with multi sums of products we construct and study certain liouville integrable superintegrable and non commutative integrable systems which are associated with multi sums of products