genus 4 trigonal reduction of the benney equations it was shown by gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 that n parameter reductions of the benney equations correspond to particular n parameter families of conformal maps in recent papers baldwin and gibbons 2003 j phys a math gen 36 8393 417 baldwin and gibbons 2004 j phys a math gen 37 5341 54 the present authors have constructed examples of such reductions where the mappings take the upper half p plane to a polygonal slit domain in the plane in those cases the mapping function was expressed in terms of the derivatives of kleinian functions of hyperelliptic curves restricted to the one dimensional stratum 1 of the divisor this was done using an extension of the method given in enolskii et al 2003 j nonlinear sci 13 157 extended to a genus 3 curve enolski v z and gibbons j addition theorems on the strata of the theta divisor of genus three hyperelliptic curves in preparation here we use similar ideas but now applied to a trigonal curve of genus 4 fundamental to this approach is a family of differential relations which satisfies on the divisor again it is shown that the mapping function is expressible in terms of quotients of derivatives of on the divisor 1 one significant by product is an expansion of the leading terms of the taylor series of for the given family of 3 5 curves to the best of the authors knowledge this is new

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

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

on the link between umbilic geodesics and soliton solutions of nonlinear pdes in this paper we describe a new class of soliton solutions called umbilic solitons for certain nonlinear integrable pdes these umbilic solitons have the property that as the space variable x tends to infinity the solution tends to a periodic wave and as x tends to minus infinity it tends to a phase shifted wave of the same shape the equations admitting solutions in this new class include the dym equation and equations in its hierarchy the methods used to find and analyse these solutions are those of algebraic and complex geometry we look for classes of solutions by constructing associated finite dimensional integrable hamiltonian systems on riemann surfaces in particular in this setting we use geodesics on n dimensional quadrics to find the spatial or x flow which together with the commuting t flow given by the equation itself defines new classes of solutions amongst these geodesics particularly interesting ones are the umbilic geodesics which then generate the class of umbilic soliton solutions this same setting also enables us to introduce another class of solutions of dym like equations which are related to elliptic and umbilic billiards

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

an algorithm for discrete tomography abstract there are many algorithms in the literature for the approximating reconstruction of a binary matrix from its line sums in this paper we provide an algorithm which starts from the line sums of an unknown binary matrix f and outputs an integer matrix s with small entries in absolute values such that the line sums of f and s coincide we also give the results of some experiments with the algorithm

reconstructing convex polyominoes from horizontal and vertical projections in 1 we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections we defined an algorithm that establishes the existence of a convex polyomino whose horizontal and vertical projections are equal to a pair of assigned vectors h v with h m and v n its computational cost is o n4m4 in this paper we introduce some operations for recontructing convex polyominoes by means of vectors h s and v s partial sums these operations allows us to define a new algorithm whose complexity is less than o n2m2

a posteriori error analysis of an augmented mixed finite element method for darcy flow abstract we develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for darcy flow the stabilized formulation is obtained by adding to the standard dual mixed approach suitable residual type terms arising from darcy s law and the mass conservation equation we derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding galerkin scheme are well posed then we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient finally we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice

on an adaptive stabilized mixed finite element method for the oseen problem with mixed boundary conditions abstract we consider the oseen problem with nonhomogeneous dirichlet boundary conditions on a part of the boundary and a neumann type boundary condition on the remaining part suitable least squares terms that arise from the constitutive law the momentum equation and the dirichlet boundary condition are added to a dual mixed formulation based on the pseudostress velocity variables we prove that the new augmented variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas elements and the velocity is approximated by continuous piecewise polynomials we develop an a posteriori error analysis based on a helmholtz type decomposition and derive a posteriori error indicators that consist of two residual terms per element except on those elements with a side on the dirichlet boundary where they both have two additional terms we prove that these a posteriori error indicators are reliable and locally efficient finally we provide several numerical experiments that support the theoretical results

binary matrices under the microscope a tomographical problem a binary matrix can be scanned by moving a fixed rectangular window sub matrix across it rather like examining it closely under a microscope with each viewing a convenient measurement is the number of 1s visible in the window which might be thought of as the luminosity of the window the rectangular scan of the binary matrix is then the collection of these luminosities presented in matrix form we show that at least in the technical case of a smoothmxn binary matrix it can be reconstructed from its rectangular scan in polynomial time in the parameters m and n where the degree of the polynomial depends on the size of the window of inspection for an arbitrary binary matrix we then extend this result by determining the entries in its rectangular scan that preclude the smoothness of the matrix

augmented mixed finite element methods for the stationary stokes equations in this paper we introduce and analyze two augmented mixed finite element methods for a velocity pressure stress formulation of the stationary stokes equations our approach which extends analog results for linear elasticity problems is based on the introduction of the galerkin least squares type terms arising from the constitutive and equilibrium equations and the dirichlet boundary condition for the velocity all of them multiplied by suitable stabilization parameters we show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms whence the associated galerkin schemes become well posed for any choice of finite element subspaces in particular we can use continuous piecewise linear velocities piecewise constant pressures and raviart thomas elements for the stresses thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation alternatively the above factor reduces to 4 when a second augmented variational formulation involving only the velocity and the stress as unknowns is employed next we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element schemes finally extensive numerical experiments illustrating the performance of the augmented mixed finite element methods confirming the properties of the a posteriori estimators and showing the behavior of the associated adaptive algorithms are reported

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

the higher times approach to multisoliton solutions of the harry dym equation an n cusp soliton solution of the harry dym equation hde is constructed explicitly the proposed approach is based on the algebraic effectivization of the reciprocal links between the systems consisting of n higher order analogues of the hde and the corresponding analogues of the kdv equation an important role in the discussed scheme belongs to the reciprocal auto backlund transformation for the dym hierarchy

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

a completely integrable hamiltonian system the dynamical system characterized by the hamiltonian h q p j k 1 n p jp kf q dj q k with f x cos x sin x is completely integrable here n is an arbitrary positive integer and are 4 arbitrary constants and real and both real or both imaginary

an a posteriori error analysis of a velocity pseudostress formulation of the generalized stokes problem abstract we consider a stabilized mixed finite element method introduced recently for the generalized stokes problem the method is obtained by adding suitable least squares terms to the dual mixed variational formulation of the problem in terms of the velocity and the pseudostress we obtain a new a posteriori error estimator of residual type and prove that it is reliable and locally efficient specifically we develop an a posteriori error analysis based on the quasi helmholtz decomposition which allows us to prove the so called local efficiency of the estimator with a non homogeneous boundary condition finally we present some numerical examples that confirm the theoretical properties of our approach

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

scanning integer matrices by means of two rectangular windows this paper deals with the reconstruction of integer matrices from rectangular scans in particular since the case of one rectangular scan has already been treated in a previous paper we consider two rectangular scans given as two integer matrices and we investigate the existence and the possibility of reconstruction of a third binary matrix which is compatible with them furthermore our inspection implies interesting side results about the number of these reconstructed matrices for different choices of the dimensions of two windows used in the input scans

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

generalized jacobians of spectral curves and completely integrable systems consider an ordinary differential equation which has a lax pair representation dot a x a x b x where a x is a matrix polynomial with a fixed regular leading coefficient and the matrix b x depends only on a x such an equation can be considered as a completely integrable complex hamiltonian system we show that the generic complex invariant manifold of this lax pair is an affine part of a non compact commutative algebraic group the generalized jacobian of the spectral curve x y p x y 0 with its points at infinity identified moreover for suitable b x the hamiltonian vector field defined by the lax pair on the generalized jacobian is translation invariant

augmented mixed finite element method for the oseen problem a priori and a posteriori error analyses abstract we propose a new augmented dual mixed method for the oseen problem based on the pseudostress velocity formulation the stabilized formulation is obtained by adding to the dual mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations we prove that for appropriate values of the stabilization parameters the new variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas or brezzi douglas marini elements and the velocity is approximated by continuous piecewise polynomials moreover we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient finally we include several numerical experiments that support the theoretical results

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

an introduction to periodical discrete sets from a tomographical perspective in this paper we introduce a new class of binary matrices whose entries show periodical configurations and we furnish a first approach to their analysis from a tomographical point of view in particular we propose a polynomial time algorithm for reconstructing matrices with a special periodical behavior from their horizontal and vertical projections we succeeded in our aim by using a reduction involving polyominoes which can be characterized by means of 2 sat formulas