a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

erratum solution of the one dimensional n body problems with quadratic and or inversely quadratic pair potentials j math phys 12 419 436 1971

a hybridizable discontinuous galerkin method for modeling fluid structure interaction abstract this work presents a novel application of the hybridizable discontinuous galerkin hdg finite element method to the multi physics simulation of coupled fluid structure interaction fsi problems recent applications of the hdg method have primarily been for single physics problems including both solids and fluids which are necessary building blocks for fsi modeling utilizing these established models hdg formulations for linear elastostatics a nonlinear elastodynamic model and arbitrary lagrangian eulerian navier stokes are derived the elasticity formulations are written in a lagrangian reference frame with the nonlinear formulation restricted to hyperelastic materials with these individual solid and fluid formulations the remaining challenge in fsi modeling is coupling together their disparate mathematics on the fluid solid interface this coupling is presented along with the resultant hdg fsi formulation verification of the component models through the method of manufactured solutions is performed and each model is shown to converge at the expected rate the individual components along with the complete fsi model are then compared to the benchmark problems proposed by turek and hron 1 the solutions from the hdg formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

analysis of hdg methods for stokes flow in this paper we analyze a hybridizable discontinuous galerkin method for numerically solving the stokes equations the method uses polynomials of degree k for all the components of the approximate solution of the gradient velocity pressure formulation the novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method it renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges as a consequence we prove that the approximations of the velocity gradient the velocity and the pressure converge with the optimal order of convergence of k 1 in l 2 for any k 0 moreover taking advantage of the superconvergence properties of the velocity we introduce a new element by element postprocessing to obtain a new velocity approximation which is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 numerical experiments are presented which validate the theoretical results

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

extending conceptualisation modes for generalised formal concept analysis formal concept analysis fca is an exploratory data analysis technique for boolean relations based on lattice theory its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes pairs of dually isomorphic sets of objects and attributes called formal concepts form a concept lattice but actually model only a conjunctive mode of conceptualisation in this paper we augment this formalism in two ways first we extend fca to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal logic motivated way this creates the three new types of concepts and lattices of extended fca viz the lattice of neighbourhood of objects that of attributes and the lattice of unrelatedness second we consider incidences with values in idempotent semirings concretely the completed max plus or schedule algebra email protected m a x and focus on generalising fca to try and replicate the modes of conceptualisation mentioned above to provide a concrete example of the use of these techniques we analyse the performance of multi class classifiers by conceptually analysing their confusion matrices

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

new finite element methods in computational fluid dynamics by h div elements in this paper the authors present two formulations for the stokes problem which make use of the existing h rm div elements of the raviart thomas type originally developed for the second order elliptic problems in addition two new h rm div elements are constructed and analyzed particularly for the new formulations optimal order error estimates are established for the corresponding finite element solutions in various sobolev norms the finite element solutions feature a full satisfaction of the continuity equation when existing raviart thomas type elements are employed in the numerical scheme

reduction of superintegrable systems the anisotropic harmonic oscillator we introduce a 2n parametric family of maximally superintegrable systems in n dimensions obtained as a reduction of an anisotropic harmonic oscillator in a 2n dimensional configuration space these systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta they generalize known examples of superintegrable models in the euclidean plane

gray scale morphological associative memories neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns thus far the emphasis in research on morphological associative memory systems has been on binary models although a number of notable features of autoassociative morphological memories amms such as optimal absolute storage capacity and one step convergence have been shown to hold in the general gray scale setting in this paper we make extensive use of minimax algebra to analyze gray scale autoassociative morphological memories specifically we provide a complete characterization of the fixed points and basins of attractions which allows us to describe the storage and recall mechanisms of gray scale amms computer simulations using gray scale images illustrate our rigorous mathematical results on the storage capacity and the noise tolerance of gray scale morphological associative memories mams finally we introduce a modified gray scale amm model that yields a fixed point which is closest to the input pattern with respect to the chebyshev distance and show how gray scale amms can be used as classifiers

an augmented mixed finite element method for the vorticity velocity pressure formulation of the stokes equations this paper deals with the numerical approximation of the stationary two dimensional stokes equations formulated in terms of vorticity velocity and pressure with non standard boundary conditions here by introducing a galerkin least squares term we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem we propose two families of mixed finite elements to solve the discrete problem in the first family the unknowns are approximated by piecewise continuous and quadratic elements brezzi douglas marini and piecewise constant finite elements respectively while in the second family the unknowns are approximated by piecewise linear and continuous raviart thomas and piecewise constant finite elements respectively the wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the babuska brezzi theory we establish a priori error estimates in the natural norms and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes additional examples of cases not covered by our theory are also presented

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

the derivation of hybridizable discontinuous galerkin methods for stokes flow in this paper we introduce a new class of discontinuous galerkin methods for the stokes equations the main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries we present four ways of hybridizing the methods which differ by the choice of the globally coupled unknowns classical methods for the stokes equations can be thought of as limiting cases of these new methods

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

the complete kepler group can be derived by lie group analysis it is shown that the complete symmetry group for the kepler problem as introduced by krause can be derived by lie group analysis the same result is true for any autonomous system

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

the determination of nonlocal symmetries by the technique of reduction of order autonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an ansatz on the detailed structure of the symmetry because the coefficient function for the original independent variable appears only as its derivative in the reduced system symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

algebra of pseudodifferential operators and symmetries of equations in the kadomtsev petviashvili hierarchy point symmetries are obtained for all equations in the kp hierarchy the lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tn the symmetry algebra is a semidirect sum of a virasoro algebra and a kac moody one the positive part of this algebra is embedded into the known w algebra of kp symmetries and into the free fermion algebra l the corresponding action on the tau function is presented the negative part of the point symmetries does not fit into the free fermion algebra but is embedded into a p algebra based on the algebra of pseudodifferential operators

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

lie symmetries and superintegrability we show that a known superintegrable system in two dimensional real euclidean space post and winternitz 2011 j phys a math theor 44 162001 can be transformed into a linear third order equation consequently we construct many autonomous integrals polynomials up to order 18 for the same system the reduction method and the connection between lie symmetries and jacobi last multiplier are used

max plus algebra based wavelet transforms and their fpga implementation for image coding in the present paper we propose new image coding methods based on max plus algebra based wavelet transforms mp wavelets max plus algebra is an algebraic structure on the set of integers endowed with maximum minimum and standard addition as operations the proposed wavelet decomposition schemes are novel classes of morphological wavelets which in contrast to the existing approaches involve only integers and only max min and addition in their computation since they do not involve floating point calculations and multiplications mp wavelets have a very low computational complexity and they are suitable for efficient hardware implementation through image coding experiments using 10 000 images extracted from low resolution web crawled misc database the characteristics of the proposed mp wavelets are presented furthermore we show a computational complexity analysis to compare mp wavelets to one of the fastest wavelet transforms the conventional haar wavelet the applicability of mp wavelets in video coding is also confirmed by image coding experiments using residual frames finally the field programmable gate arrays fpga implementation of mp wavelets is presented to confirm suitability of mp wavelets for hardware implementation

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

morphological perceptrons with competitive learning lattice theoretical framework and constructive learning algorithm a morphological neural network is generally defined as a type of artificial neural network that performs an elementary operation of mathematical morphology at every node possibly followed by the application of an activation function the underlying framework of mathematical morphology can be found in lattice theory with the advent of granular computing lattice based neurocomputing models such as morphological neural networks and fuzzy lattice neurocomputing models are becoming increasingly important since many information granules such as fuzzy sets and their extensions intervals and rough sets are lattice ordered in this paper we present the lattice theoretical background and the learning algorithms for morphological perceptrons with competitive learning which arise by incorporating a winner take all output layer into the original morphological perceptron model several well known classification problems that are available on the internet are used to compare our new model with a range of classifiers such as conventional multi layer perceptrons fuzzy lattice neurocomputing models k nearest neighbors and decision trees

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method

the spectra of irreducible matrices over completed idempotent semifields motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally ordered semirings or dioids we find non null eigenvectors for every non null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields