linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

hp discontinuous galerkin method based on local higher order reconstruction we present a new adaptive higher order finite element method hp fem for the solution of boundary value problems formulated in terms of partial differential equations pdes the method does not use any information about the problem to be solved which makes it robust and equation independent it employs a higher order reconstruction scheme over local element patches which makes it faster and easier to parallelize compared to hp adaptive methods that are based on the solution of a reference problem on a globally hp refined mesh the method can be used for the solution of linear as well as nonlinear problems discretized by conforming or non conforming finite element methods and it can be combined with arbitrary a posteriori error estimators the performance of the method is demonstrated by several examples carried out by the discontinuous galerkin method

creating and relating three dimensional integrable maps we show how some integrable third order difference equations recently given in the literature are related to one another by the process of interchanging parameters and integrals using the same process we then create a 21 parameter family of integrable third order difference equations that contains the previous examples as special cases our methodology illustrates that the combination of finding 2 integrals i e integrals of the second iterate of the map exploiting linear parameter dependence and using the interchange process provides a powerful way to relate and create higher dimensional discrete integrable systems

a collection of 2d elliptic problems for testing adaptive grid refinement algorithms adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations pdes the development of new algorithms and computer codes for the solution of pdes usually involves the use of proof of concept test problems 2d elliptic problems are often used as the first test bed for new algorithms and codes this paper contains a set of twelve parametrized 2d elliptic test problems for adaptive grid refinement algorithms and codes the problems exhibit a variety of types of singularities near singularities and other difficulties

interchanging parameters and integrals in dynamical systems the mapping case we consider dynamical systems with discrete time maps that possess one or more integrals depending upon parameters we show that integrals can be used to replace parameters in the original map so as to construct a different map with different integrals we also highlight a process of reparametrization that can be used to increase the number of parameters in the original map prior to using integrals to replace them properties of the original map and the new map are compared the theory is motivated by and illustrated with examples of a three dimensional trace map and some four dimensional maps previously shown to be integrable

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

a new family of four dimensional symplectic and integrable mappings abstract we investigate the generalisations of the quispel roberts and thompson qrt family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables we assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with poisson brackets of the weyl type all mappings satisfying these conditions are shown to be integrable either as four dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double discrete versions of the modified korteweg devries mkdv and sine gordon sg equations or by reduction to two dimensional mappings with one integral of the symmetric qrt family

on some classes of discrete polynomials and ordinary difference equations we introduce two classes of discrete polynomials and use them to construct ordinary difference equations admitting a lax representation in terms of these polynomials we also construct lattice integrable hierarchies in their explicit form and show some examples

an a posteriori error estimate for hp adaptive dg methods for convection diffusion problems on anisotropically refined meshes we prove an a posteriori error estimate for hp adaptive discontinuous galerkin methods for the numerical solution of convection diffusion equations on anisotropically refined rectangular elements the estimate yields global upper and lower bounds of the errors measured in terms of a natural norm associated with diffusion and a semi norm associated with convection the anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure we present a series of numerical experiments to test the feasibility of this approach within a fully automated hp adaptive refinement algorithm

on integrals for some class of ordinary difference equations admitting a lax representation we consider two infinite classes of ordinary difference equations admitting lax pair representation discrete equations in these classes are parameterized by two integers and we describe the first integrals for these two classes in terms of special discrete polynomials we show an equivalence between two difference equations belonging to different classes corresponding to the same pair k s we show that solution spaces of different ordinary difference equations with a fixed value of s k are organized in a chain of inclusions

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

integrable fourth order difference equations in this paper an attempt is made to find four dimensional analogs of two dimensional quispel roberts and thompson mappings and identified four distinct cases have been identified the obtained mappings are measure preserving the integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible

third order integrable difference equations generated by a pair of second order equations we show that the third order difference equations proposed by hirota kimura and yahagi are generated by a pair of second order difference equations in some cases the pair of the second order equations are equivalent to the quispel robert thomson qrt system but in the other cases they are irrelevant to the qrt system we also discuss an ultradiscretization of the equations

bilinear trilinear forms and exact solution of certain fourth order integrable difference equations a systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation x n 4 f x n x n 1 x n 2 x n 3 or xn 4 f xn xn 1 xn 2 xn 3 is made as an illustration we consider fourth order symplectic integrable difference equations reported by capel and sahadevan physica a 289 86 2001 and derived their bilinear or trilinear forms also it is shown that the obtained bilinear representations admit exact solution of rational form

higher dimensional integrable mappings we construct higher dimensional analogues of two dimensional integrable mappings preserving one biquadratic integral we also provide their non autonomous forms

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

sufficient number of integrals for the pth order lyness equation in this communication we present a sufficient number of first integrals for the lyness equation of arbitrary order we first use the staircase method quispel et al 1991 physica a 173 243 66 to construct integrals of a derivative equation of the lyness equation closed form expressions for the integrals are given based on a non commutative vieta expansion the integrals of the lyness equation then follow directly from these integrals previously found integrals for the lyness equation arise as special cases of our new set of integrals

hp optimization of finite element approximations analysis of the optimal mesh sequences in one dimension this paper addresses the problem of the construction of the optimal hp meshes for the singular solution x 12 in the one dimensional setting first a review of the asymptotic analysis of gui and babuska 1 is given and is used to characterize the optimal hp meshes which are geometrically graded toward the singular point the asymptotically optimal hp meshes are compared with the hp meshes which are constructed by employing the hp strategy of rachowicz et al 2 which employs the principle of maximum change of error per change in the number of degrees of freedom and the true optimal hp meshes which minimize the error for fixed number of degrees of freedom and which are determined by searching through the entire sequence of the admissible hp meshes it is shown that for the singular solution the meshes constructed by the hp strategy of rachowicz coincide with the true optimal meshes nevertheless there exist various solutions for which the algorithm which employs the principle of maximum change of error may not converge for this reason the hp strategy of rachowicz is combined with the classical feedback algorithm to obtain a robust hp strategy which always converges further it is shown that the asymptotically optimal hp meshes are very close to the true optimal hp meshes in the entire range of practical computation the effect of the accuracy of the error estimator on the optimality of the constructed hp meshes is also analyzed it is shown that unless the estimator is sufficiently accurate in the element adjacent to the singular point the meshes constructed by the adaptive hp strategy can differ substantially from the true optimal meshes

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

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

spectral theory of certain unbounded jacobi matrices abstract using the conjugate operator method of mourre we study the spectral theory of a class of unbounded jacobi matrices we especially focus on the case where the off diagonal entries a n n 1 o 1 and diagonal ones b n n 1 o 1 with 0 r

the discrete chazy iii system of labrunie conte is not integrable we analyse the discrete form of the chazy iii equation proposed by labrunie and conte with the help of two different integrability criteria singularity confinement and algebraic entropy we show that for all values of the free parameter this third order mapping fails both criteria and thus cannot be integrable

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

multidimensional maps of qrt type we consider a class of multidimensional maps which naturally generalize the qrt map of the plane our 2n dimensional maps are volume preserving and have n rational invariants but we do not generally have a symplectic form however many specializations and reductions are integrable some of which we present included in these are some new four dimensional generalizations of the mcmillan map

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

some properties of the k dimensional lyness map this paper is devoted to study some properties of the k dimensional lyness map f x1 xk x2 xk a ki 2xi x1 our main result presents a rational vector field that gives a lie symmetry for f this vector field is used for k 5 to give information about the nature of the invariant sets under f when k is odd we also present a new as far as we know first integral for ff which allows us to deduce in a very simple way several properties of the dynamical system generated by f in particular for this case we prove that except on a given codimension one algebraic set none of the positive initial conditions can be a periodic point of odd period

closed form expressions for integrals of mkdv and sine gordon maps we present closed form expressions for approximately n integrals of 2n dimensional maps the maps are obtained by travelling wave reductions of the modified korteweg de vries equation and of the sine gordon equation respectively we provide the integrating factors corresponding to the integrals moreover we show how the integrals and the integrating factors relate to the staircase method

analysis of the discontinuous galerkin method for nonlinear convection diffusion problems abstract the subject matter is the analysis of the discontinuous galerkin finite element method applied to a nonlinear convection diffusion problem in the contrary to the standard fem the requirement of the conforming properties is omitted this allows us to consider general polyhedral elements with mutually disjoint interiors we do not require their convexity but assume only that they are star shaped we present an error analysis for the case of a nonsymmetric discretization of diffusion terms theoretical results are accompanied by numerical experiments

spectral properties of some jacobi matrices with double weights abstract we study the spectral properties of jacobi matrices with the weights satisfying 2n 1 2n na or 2n 2n 1 na a 0 we show that for a 1 these are cases of spectral phase transitions in a we use a new method of estimating transfer matrix products to describe the absolutely continuous part of these operators for a 1 the existence of a spectral gap is proved we also show how the results for double weights can be used for the spectral analysis of the jacobi matrices related to some birth and death processes previously studied by janas and naboko

integrable mappings of the plane preserving biquadratic invariant curves we provide a general framework to construct integrable mappings of the plane that preserve a one parameter family b x y k of biquadratic invariant curves where parametrization by k is very general these mappings are reversible by construction i e they are the composition of two involutions and can be shown to be measure preserving they generalize integrable maps previously given by mcmillan and quispel roberts and thompson by considering a transformation of the case of the symmetric biquadratic to a canonical form we provide a normal form for the symmetric integrable map acting on each invariant curve we give a lax pair for a large subclass of our symmetric integrable maps including at least a 10 parameter subfamily of the 12 parameter symmetric quispel roberts thompson maps

duality for discrete integrable systems a new class of discrete dynamical systems is introduced via a duality relation for discrete dynamical systems with a number of explicitly known integrals the dual equation can be defined via the difference of an arbitrary linear combination of integrals and its upshifted version we give an example of an integrable mapping with two parameters and four integrals leading to a four dimensional dual mapping with four parameters and two integrals we also consider a more general class of higher dimensional mappings arising via a travelling wave reduction from the integrable mkdv partial difference equation by differencing the trace of the monodromy matrix we obtain a class of novel dual mappings which is shown to be integrable as level set dependent versions of the original ones

analysis and application of the iipg method to quasilinear nonstationary convection diffusion problems we develop a numerical method for the solution of convection diffusion problems with a nonlinear convection and a quasilinear diffusion we employ the so called incomplete interior penalty galerkin iipg method which is suitable for a discretization of quasilinear diffusive terms we analyse a use of the iipg technique for a model scalar time dependent convection diffusion equation and derive hp a priori error estimates in the l 2 norm and the h 1 seminorm moreover a set of numerical examples verifying the theoretical results is performed finally we present a preliminary application of the iipg method to the system of the compressible navier stokes equations

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

arbitrary level hanging nodes for adaptive hp fem approximations in 3d in this paper we discuss constrained approximation with arbitrary level hanging nodes in adaptive higher order finite element methods hp fem for three dimensional problems this technique enables using highly irregular meshes and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh the technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes we present a detailed mathematical description of the method and illustrate it with numerical examples

a commutator approach to absolute continuity for unbounded jacobi operators this paper uses commutator equations to study the absolute continuity of spectral measures associated with certain subclasses of unbounded self adjoint jacobi matrix operators determined by properties of the diagonal and subdiagonal sequences if the diagonal sequence is the zero sequence properties of the difference sequence of the subdiagonal determine the choice of a bounded operator for the commutator equation the structure of the resulting commutator leads to results on absolute continuity

a novel nth order difference equation that may be integrable we derive an nth order difference equation as a dual of a very simple periodic equation and construct n 1 2 explicit integrals and integrating factors of this equation in terms of multi sums of products we also present a generating function for the degrees of its iterates exhibiting polynomial growth in conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of levi and yamilov these three facts combine to suggest the integrability of the nth order difference equation

super integrable four dimensional autonomous mappings a systematic investigation of the complete integrability of a fourth order autonomous difference equation of the type w n 4 w n f w n 1 w n 2 w n 3 is presented we identify seven distinct families of four dimensional mappings which are super integrable and have three independent integrals via a duality relation as introduced in a recent paper by quispel capel and roberts 2005 j phys a math gen 38 3965 80 it is observed that these seven families can be related to the four dimensional symplectic mappings with two integrals including all the four dimensional periodic reductions of the integrable double discrete modified korteweg devries and sine gordon equations treated in an earlier paper by two of us capel and sahadevan 2001 physica a 289 86 106

third order difference equations with two rational integrals a systematic investigation to derive three dimensional analogs of two dimensional quispel roberts and thompson qrt mappings is presented the question of integrability of the obtained three dimensional mappings with two independent integrals is also analyzed it is also shown that there exist three dimensional qrt maps with three n dependent integrals

existence of periodic solutions for a 2nth order nonlinear difference equation abstract the authors consider the 2 n th order difference equation n r t n n x t n f t x t 0 n z 3 t z where f z r r is a continuous function in the second variable f t t z f t z for all t z z r r t t r t for all t z and t a given positive integer by the linking theorem some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation

goal oriented hp adaptivity for elliptic problems we propose and test a fully automatic goal oriented hp adaptive strategy for elliptic problems the method combines two techniques the standard goal oriented adaptivity based on a simultaneous solution of a dual problem and a recently proposed hp strategy based on minimizing the projection based interpolation error of a reference solution the proposed strategy is illustrated with two numerical examples laplace equation in l shape domain and an axisymmetric maxwell problem involving radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

sufficient conditions for dynamical systems to have pre symplectic or pre implectic structures abstract we present a number of alternative sufficient conditions for the existence of pre symplectic or pre implectic poisson structures for both ordinary differential ode and ordinary difference o e equations four alternative sets of conditions are obtained for odes and o es in n dependent variables 1 a vector field in involution with the ode and an integral or two symmetries for an o e imply a pre implectic structure 2 volume preservation and n 2 symmetries imply a pre symplectic structure 3 volume preservation and n 2 integrals imply a pre implectic structure 4 complex implectic structure implies infinitely many real implectic structures in all but the first case the methods can give a set of distinct structures

a goal oriented anisotropic hp mesh adaptation method for linear convection diffusion reaction problems abstract we deal with the numerical solution of linear convection diffusion reaction equations using the h p variant of the discontinuous galerkin method on triangular grids we develop a mesh adaptive algorithm which modifies the size and shape of mesh elements and the corresponding polynomial approximation degrees in order to decrease the error in a target functional under the given tolerance we recall some theoretical results and describe in detail several technical aspects of this approach numerical experiments demonstrating the accuracy efficiency and robustness of the algorithm are presented

on the conservation laws for nonlinear partial difference equations in this paper a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations p es the identified p es can be classified into qrt type kdv type mkdv type and sg type the integrability of the identified partial difference equations is also discussed

hp adaptation driven by polynomial degree robust a posteriori error estimates for elliptic problems we devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions the adaptation is based on equilibrated flux estimates these estimates are presented here for inhomogeneous dirichlet and neumann boundary conditions for spatially varying polynomial degree and for mixed rectangular triangular grids possibly containing hanging nodes they deliver a global error upper bound with constant one and up to data oscillation error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound we numerically asses the estimates and several hp adaptive strategies using the interior penalty discontinuous galerkin method asymptotic exactness is observed for all the symmetric nonsymmetric odd degrees and incomplete variants on non nested unstructured triangular grids for a smooth solution and uniform refinement exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement

an hp adaptive strategy based on continuous sobolev embeddings a new hp adaptive approach for finite element methods is presented specifically the paper addresses the well known question of how to effectively decide between h and p refinement within an adaptive hp finite element procedure the main idea is based on testing the smoothness of the underlying pde solution by means of suitable continuous sobolev embeddings numerical experiments confirm the effectiveness of the proposed method

a comparison of refinement indicators for p adaptive simulations of steady and unsteady flows using discontinuous galerkin methods abstract this paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p adaptive algorithms for discontinuous galerkin dg methods residual error discretization error and feature based indicators are compared by studying their effect on convergence history computational gain and refinement regions selected by the adaptive algorithm the analysis is initially carried out on steady flow configurations static p adaptive simulations of the periodic flow past a cylinder at r e 100 are then performed compared to uniform p refinement a reduction between 50 and 75 in the number of degrees of freedom is obtained for all test cases considered the accuracy and efficiency observed for the small scale energy density 1 and spectral decay 2 indicators demonstrate their great potential for the adaptive simulation of unsteady flows

absolute continuity for unbounded jacobi matrices with constant row sums we investigate the spectral properties of a class of jacobi matrices in which the subdiagonal entries are quadratics and the row sums are constants

closed form expressions for integrals of traveling wave reductions of integrable lattice equations we give a method to calculate closed form expressions in terms of multi sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations important ingredients are the staircase method a non commutative vieta formula and certain splittings of the lax matrices the method is applied to all equations of the adler bobenko suris classification with the exception of q4

higher dimensional integrable mappings derived from coupled discrete nonlinear schrodinger equations a systematic investigation to derive higher dimensional coupled autonomous integrable mappings from scalar and coupled discrete nonlinear schordinger equations is presented the integrability in the sense of liouville of the obtained higher dimensional mappings is ensured by showing that each of them is symplectic and possesses sufficient number of integrals of motion

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