postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

pseudospectral method and darvishi s preconditioning for solving system of time dependent partial differential equations abstract in this paper we solve system of time dependent partial differential equations pdes by using pseudospectral method and darvishi s preconditioning to reduce round off error in spectral collocation method we use darvishi s preconditioning firstly theory of application of spectral collocation method on system of time dependent partial differential equations presented this method yields a system of ordinary differential equations odes secondly we use fourth order runge kutta formula for the numerical integration of the system of ode we consider some examples to illustrate the performance of the method described

realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

a numerical solution for non classical parabolic problem based on chebyshev spectral collocation method a numerical method based on chebyshev polynomials and local interpolating functions is proposed for solving the one dimensional parabolic partial differential equation subject to non classical conditions to assess its validity and accuracy the method is applied to solve several test problem

descriptor representations without direct feedthrough term abstract descriptor representations are considered that are given by e a b c d with d 0 minimality under external equivalence is characterized in terms of the matrices e a b and c also transformations are given by which minimal e a b c representations are related under external equivalence the transformations turn out to be more simple than in the d 0 case algorithms for rewriting an e a b c d representation in e a b c form are also given finally a realization procedure is presented for obtaining a minimal e a b c representation for a system that is given in polynomial matrix fractional form

a petrov galerkin spectral element technique for heterogeneous porous media flow abstract a novel spectral element technique is examined in which the test functions are different from the approximating elements examples are given for a simple 1 d helmholtz equation burger s equation with a small viscosity and for darcy s equation with a discontinuous hydraulic conductivity

minimality of descriptor representations under external equivalence abstract necessary and sufficient conditions are derived for the minimality of descriptor representations under external equivalence these conditions are stated completely in terms of the matrices e a b c and d use is made of the close connection between the descriptor representation and the so called pencil representation this connection is further exploited to derive the transformation group for minimal descriptor representations it is shown that the transformations coincide with the operations of strong equivalence as introduced by verghese et al ieee trans aut control ac 26 811 831 1981

a modified chebyshev pseudospectral dd algorithm for the gbh equation in this paper a chebyshev spectral collocation domain decomposition dd semi discretization by using a grid mapping derived by kosloff and tal ezer in space is applied to the numerical solution of the generalized burger s huxley gbh equation to reduce roundoff error in computing derivatives we use the above mentioned grid mapping in this work we compose the chebyshev spectral collocation domain decomposition and kosloff and tal ezer grid mapping elaborately firstly the theory of application of the chebyshev spectral collocation method with grid mapping and dd on the gbh equation is presented this method yields a system of ordinary differential algebraic equations daes secondly we use a fourth order runge kutta formula for the numerical integration of the system of daes application of this modified method to the gbh equation show that this method m dd is faster and more accurate than the standard chebyshev spectral collocation dd s dd method

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

the elementary divisors of incidence matrices between certain subspaces of a finite symplectic space abstract the elementary divisors of the incidence matrices between 1 dimensional subspaces and r dimensional isotropic subspaces of a finite symplectic space are computed

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

input output structure of linear differential algebraic systems the problem of describing the input output structure of general linear differential algebraic systems is addressed explicit formulas are given in terms of the original parameters for systems with an arbitrary amount of redundancy these formulas allow one to establish whether the system determines an input output relation at all and if so they describe the rank of the transfer matrix and its pole zero structure at infinity the formulas may be seen as generalizations of a number of classical results on the input output structure of standard state space systems and descriptor systems satisfying certain constraints for the derivation the pencil representation rather than the descriptor representation is used

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

on extended weil representation for finite general linear group and howe correspondence abstract let q be a power of a prime number we construct a variety over a finite field f q and show that its middle cohomology realizes the heisenberg weil representations of a general linear group g l n q for any integer n 1 if n is even by using an endomorphism of this variety we can construct a representation of a semidirect group g l n q z 2 z which is an extension of the weil representation of g l n q by inflating this by a natural homomorphism s p n q o 2 q g l n q z 2 z we obtain a representation of s p n q o 2 q this representation can be used for the howe correspondence for s p n o 2 in any characteristic as application we show that unipotency is preserved under the correspondence for s p n o 2

application of asymptotic waveform evaluation to hybrid fe bi mlfma for fast rcs computation over a frequency band the asymptotic waveform evaluation awe technique has been widely used in the method of moment mom the finite element method fem and the hybrid finite element boundary integral method fe bi for fast computation over a wide frequency band however how to apply awe to the multilevel fast multipole algorithm mlfma still remains challenge this paper proposes an efficient and robust approach to apply awe to the hybrid finite element boundary integral multilevel fast multipole algorithm fe bi mlfma the validity of the proposed approach is verified by numerical examples for scattering problems numerical experiments show that the application of awe to fe bi mlfma can greatly improve the efficiency of fe bi mlfma for computation of scattering over a wide frequency band

fourier collocation splittings for partial differential equations in this article we study computational issues related to a nonlinear galerkin type splitting nlg of partial differential equations in the case of a fourier collocation discretization we present an extension of the method to two dimensional problems and show that the sole separation of modes in nlg can bring precision and computational costs advantages to the standard collocation scheme numerical experiments with the burgers and a reaction diffusion equation for 1 and 2 dimensions are also shown

bifurcation computations on an approximate inertial manifold for the 2d navier stokes equations abstract we compare the minimum number of modes needed to capture certain bifurcations of the 2d navier stokes equations by a certain nonlinear galerkin method and by the traditional galerkin method the boundary conditions are periodic and the forcing is in a single mode two different choices of the forcing mode are tested we introduce and implement a homotopy which allows continuation between approximations of different size

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

state representations of arma models a state representation of an arbitrary arma model is computed explicitly it is shown then that every arma model is homotopy equivalent to its state representation and that two state models are homotopy equivalent if and only if they are similar

the structure of rank 3 permutation modules for o2n 2 and um 2 acting on nonsingular points abstract we study the odd characteristic structure of permutation modules for the rank 3 natural actions of o 2 n 2 n 3 and u m 2 m 4 on nonsingular points of their standard modules

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

the modulo 2 structure of rank 3 permutation modules for odd characteristic symplectic groups this paper studies the permutation representation of the symplectic group sp 2m fq where q is odd on the 1 spaces of its natural module the complete submodule lattice for the modulo l reduction of this permutation module is known for all odd primes l not dividing q in this paper we determine the complete submodule lattice for the mod2 reduction similar results are then obtained for the orthogonal group o 5 fq

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

the steady states of the kuramoto sivashinsky equation abstract the kuramoto sivashinsky equation is one of the simple nonlinear partial differential equations it represents a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation the steady states and their stabilities give a foundation for understanding the dynamics of the system they are distinguished according to their energy levels and fall into different branches that are identified by their harmonic content various types of solution include laminar states n cell states long wave distorbed n cell states the giant states and steady traveling solutions the linear stability around the steady states gives a guide to the origin of additional steady states through bifurcation and to the identification of each steady state the index rule is also useful conjectured form the numerical observation the parameter dependence of the steady traveling waves is estimated in the limit of large values of the parameter in addition a singular perturbation calculation of traveling waves near bifurcation is carried out the dynamics of a system can be guided by the systematic study of the bifurcation and stability for example if all the stationary and periodic solutions are unstable for some parameter range chaotic behavior of the system is expected since any orbit governed by the equation has to wander around forever in the phase space the orbits will then lie on strange attractors

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

adaptive pseudospectral solution of a diffuse interface model a diffuse interface type model using an energy based variational formulation with a free energy that is a function of the density and its gradients is presented all of the boundary terms are retained and related to external surface forces which can be of particular interest when considering the fluid fluid solid region the numerical solution of these types of problems can be troublesome if a thin transition layer is desired here chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied a mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived in order to achieve high precision for problems exhibiting thin transition layers a modified version of the arc length monitor function is proposed which yields a sufficiently smooth coordinate transformation at every step of the continuation process a fixed number of iterations is implemented so that the equidistribution equations are not solved completely at each step which saves a considerable amount of computational effort numerical results for the static phase field model exhibiting thin transition layers are presented

cross characteristic representations of odd characteristic symplectic groups and unitary groups in ls landazuri and seitz gave lower bounds for irreducible representations of chevalley groups in non defining characteristic when referring to irreducible representations for quasisimple groups g we will assume that the modules are nontrivial on f g see also sz gpps ho for some improvements on these bounds these results have proved to be useful in many applications in particular they have been used to classify the maximal subgroups of classical groups containing an element of prime order acting irreducibly on a subspace of large dimension cf gpps and to show that low dimensional modules in characteristic p for groups with no normal p subgroup are semisimple see gu it is also important to identify the modules which have dimension close to the smallest possible dimension and to prove that there are no irreducible modules with dimension in a certain range above it this was done in gpps and gt for sln q further improvements were obtained by brundan and kleshchev brk hiss and malle hm have obtained results similar to gt for unitary groups in this paper we consider the groups g sp2n q with n 2 and q p odd and g sun q with n 3 landazuri and seitz ls had already shown that the minimal dimension d of any irreducible module in the nondefining characteristic has dimension exactly q 1 2 for the symplectic case and q 1 q 1 for the unitary case it was proved in gpps that aside from some small exceptions every irreducible kg module in a nondefining characteristic has dimension d d 1 or at least dimension 2d in characteristic 0 tiep and zalesskii tz1 using deligne lusztig

an approximate inertial manifold for computing burgers equation abstract we present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals our algorithm is motivated from the dynamical systems point of view in particular we adapt the methodology of approximate inertial manifolds to a finite difference scheme this leads to a differential treatment in which the higher i e unresolved modes are expressed in terms of the lower modes as a particular example we derive an approximate inertial manifold for burgers equation and develop a numerical algorithm suitable for computing we perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme for all values of the parameters which are the coefficient of viscosity and the cell size we obtain a decrease in the numerical error by at least a factor of 2 0 with the modified scheme the decrease in error is substantially greater over large regions of the parameters space

a fast chebyshev method for simulating flexible wing propulsion abstract we develop a highly efficient numerical method to simulate small amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid we allow the wing s elastic modulus and mass density to vary arbitrarily with an eye towards optimizing these distributions for propulsive performance the method to determine the wing kinematics is based on chebyshev collocation of the 1d beam equation as coupled to the surrounding 2d fluid flow through small amplitude analysis of the euler equations with trailing edge vortex shedding the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1d wing kinematics a class of semi analytical solutions permits fast evaluation of this operator with o n log n operations where n is the number of collocation points on the wing this is in contrast to the minimum o n 2 cost of a direct 2d fluid solver the coupled wing fluid problem is thus recast as a pde with nonlocal operator which we solve using a preconditioned iterative method these techniques yield a solver of near optimal complexity o n log n allowing one to rapidly search the infinite dimensional parameter space of all possible material distributions and even perform optimization over this space

continuity of singular perturbations in the graph topology abstract for a certain model for singular perturbations in control systems which we motivate by a simple example we show that under weak assumptions continuity in the graph topology holds as the perturbation parameter tends to zero this may be contrasted with a result by cobb who considered a different model for singular perturbations and who found a strong condition to be necessary for continuity in that model our proof techniques are based on the characterization due to qiu and davison of the graph topology as a topology of uniform convergence

spectral collocation method for the solution of the generalized burger fisher equation abstract in this paper we present a new method for solving the generalized burger fisher equation by using the collocation formula for calculating spectral differentiation matrix for chebyshev gauss lobatto point firstly theory of application of pseudospectral method on burger fisher equations is presented this method yields burger fisher equation to a system of ordinary differential equations odes secondly we use forth order runge kutta formula for the numerical integration of the system of ode the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

an error minimized pseudospectral penalty direct poisson solver this paper presents a direct poisson solver based on an error minimized chebyshev pseudospectral penalty formulation for problems defined on rectangular domains in this study the penalty parameters are determined analytically such that the discrete l 2 error is minimized numerical experiments are conducted and the results show that the penalty scheme computes numerical solutions with better accuracy compared to the traditional approach with boundary conditions enforced strongly

a numerical solution of the generalized burger s huxley equation by spectral collocation method abstract in this paper we present a new method for solving of the generalized burger s huxley equation by using the collocation formula for calculating spectral differentiation matrix for chebyshev gauss lobatto points firstly theory of application of pseudospectral method on burger s huxley equations presented this method yields burger s huxley equation to a system of ordinary differential equations odes secondly we use fourth order runge kutta formula for the numerical integration of the system of ode the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method

chebyshev collocation spectral lattice boltzmann method for simulation of low speed flows in this study the chebyshev collocation spectral lattice boltzmann method ccslbm is developed and assessed for the computation of low speed flows both steady and unsteady flows are considered here the discrete boltzmann equation with the bhatnagar gross krook approximation based on the pressure distribution function is considered and the space discretization is performed by the chebyshev collocation spectral method to achieve a highly accurate flow solver to provide accurate unsteady solutions the time integration of the temporal term in the lattice boltzmann equation is made by the fourth order runge kutta scheme to achieve numerical stability and accuracy physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used an iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows the main advantage of using the ccslbm over other high order accurate lattice boltzmann method lbm based flow solvers is the decay of the error at exponential rather than at polynomial rates note also that the ccslbm applied does not need any numerical dissipation or filtering for the solution to be stable leading to highly accurate solutions three two dimensional 2d test cases are simulated herein that are a regularized cavity the taylor vortex problem and doubly periodic shear layers the results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement the computational efficiency of the proposed solution methodology based on the ccslbm is also examined by comparison with those of the standard streaming collision classical lbm and two finite difference lbm solvers the study indicates that the ccslbm provides more accurate and efficient solutions than these lbm solvers in terms of cpu and memory usage and an exponential convergence is achieved rather than polynomial rates the solution methodology proposed the ccslbm is also extended to three dimensions and a 3d regularized cavity is simulated the corresponding results are presented and validated indications are that the ccslbm developed and applied herein is robust efficient and accurate for computing 2d and 3d low speed flows note also that high accuracy solutions obtained by applying the ccslbm can be used as benchmark solutions for the assessment of other lbm based flow solvers

numerical simulation for the space fractional diffusion equations abstract this paper presents a novel semi analytical algorithm based on the chebyshev collocation method for the solution of space fractional diffusion equations the original fractional equation is transformed into a system of ordinary differential equations odes by the chebyshev collocation method a new semi analytical method is then used to approximate the solution of the resulting system to emphasize the reliability of the new scheme a convergence analysis is presented it is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased by presenting some numerical examples we show that the proposed method is a powerful and reliable algorithm for solving space fractional diffusion equations and can be extended to solve another space fractional partial differential equations comparison with other methods in the literature demonstrates that the proposed method is both efficient and accurate

time marching multilevel techniques for evolutionary dissipative problems in this article we use a pseudospectral fourier discretization in conjunction with a multilevel splitting of high and low modes to solve dissipative partial differential equations we develop unconditionally stable explicit techniques for the temporal integration of the linear terms and apply them to the high modes equation improving the overall temporal stability of the multilevel method and resulting in a competitive fully explicit numerical scheme for nonlinear problems in the cases where the linear term determines the time step restriction numerical experiments with the burgers equation in one and two dimensions showed substantial cpu cost reduction when comparing the resulting method with the standard spectral collocation associated with regular temporal integration schemes

accuracy and speed in computing the chebyshev collocation derivative we study several algorithms for computing the chebyshev spectral derivative and compare their roundoff error for a large number of collocation points the elements of the chebyshev differentiation matrix if constructed in the usual way are not computed accurately a subtle cause is found to account for the poor accuracy when computing the derivative by the matrix vector multiplication method methods for accurately computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately the roundoff error of the matrix vector multiplication is as small as that of the transform recursion algorithm furthermore results of the cpu time usage are shown for several different algorithms for computing the derivative by the chebyshev collocation method for a wide variety of two dimensional grid sizes on both an ibm mainframe and a cray 2 computer we find that which algorithm is fastest on a particular machine depends not only on the grid size but also on small

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

dynamics and synchronization of numerical solutions of the burgers equation for chebyshev spectral solutions of the forced burgers equation with low values of the viscosity coefficient several bifurcations and stable attractors can be observed periodic orbits quasiperiodic and strange ones may arise bistability can also be observed necessary conditions for these attractors to appear are discussed and justification for the non emerging of bistability for an example of a system symmetry break is presented as an application for the dynamical behavior of spectral solutions of burgers equation the dynamics and synchronization of unidirectionally coupling of chebyshev spectral solutions of burgers equations by means of a linear coupling are described and discussed also a nonlinear coupling is proposed and discussed

on the question of turbulence modeling by approximate inertial manifolds and the nonlinear galerkin method this paper contributes to the discussion about the ability of the aim method to model turbulent flow and about the theoretical potential of the aim ngm to provide a computational basis for the calculation of turbulent flow

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

rank 3 permutation modules of the finite classical groups abstract the cross characteristic permutation modules for the actions of the finite classical groups on singular 1 spaces of their natural modules are studied the composition factors and submodule lattices are determined

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

realization of autoregressive equations in pencil and descriptor form a linear system described by autoregressive equations with a given input output structure cannot be transformed to standard state space form if the implied input output relation is nonproper instead a realization in descriptor form must be used in this paper it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies and without going through a reduction process external equivalence is used so that even situations in which there is no transfer matrix can be considered the approach is based on the so called pencil representation of linear systems and it is shown that there is a natural realization of autoregressive equations in pencil form in this way the link between the realization theories of willems and fuhrmann can also be clarified

a linearisation method for non linear singular boundary value problems this paper presents a novel numerical approach for the solution of a class of nonlinear singular boundary value problems arising in physiology the approach is based on a new application of the successive linearisation method slm three illustrative examples are presented to demonstrate the effectiveness of the proposed method the new approach is found to give accurate results comparable to results in the literature found using existing numerical methods

system order and structure indices of linear systems in polynomial form linear time invariant finite dimensional discrete time systems are very often specified in a polynomial form representation in either a forward or backward shift operator sometimes called mfd and arma form in this paper it is shown that there exists a unifying theory for the determination of mcmillan degree and kronecker observability indices of systems represented by polynomial matrices in the two shift operators considering the mfd and arma forms as special cases treating dynamical systems in terms of their behaviour i e the set of admissible signal trajectories the notions of mcmillan degree and observability indices can be generalized to non controllable as well as to non causal systems

representability of impulsive input output behaviors in descriptor form abstract it is known that the descriptor representation of linear smooth behaviors imposes no constraints on the partitioning of external variables into inputs and outputs the issue is reconsidered here in the setting of impulsive behaviors in this context there are two different descriptor representations that may be used the conventional form and the unconventional one it is shown that the conventional descriptor form again does not constrain the choice of inputs and outputs but the unconventional form does the latter representation only allows input output partitionings in which inputs are dirac free this notion is newly defined here

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

cross characteristic representations of even characteristic symplectic groups we classify the small irreducible representations of sp 2n q with q even in odd characteristic this improves even the known results for complex representations the smallest representation for this group is much larger than in the case when q is odd this makes the problem much more difficult

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

the equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions abstract necessary and sufficient conditions for minimality of descriptor representations of impulsive smooth behaviors are derived we obtain a complete set of transformations by which minimal descriptor representations that give rise to the same behavior can be transformed into each other in particular this leads to a jump behavioral interpretation of the notion of strong equivalence of descriptor representations

state space approach to behavioral systems theory the dirac bergmann algorithm this paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised this representation is a first order hybrid representation of behaviors called pencil representation an algorithm well known after dirac and bergmann db is shown to be central in obtaining a constraint free and observable cfo state space representation of a behavior results and criteria for asymptotic stability controllability inclusions and markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation

singular state models and singular ltid systems in this paper we offer a systematic development of state representation theory for singular linear systems in the behavioural setting of willems

impulsive smooth behavior in multimode systems part i state space and polynomial representations abstract a switched or multimode system is one that can switch between various modes of operation we consider here switched systems in which the modes of operation are characterized as linear finite dimensional systems not necessarily all of the same mcmillan degree when a switch occurs from one of the modes to another of lower mcmillan degree the state space collapses and an impulse may result followed by a smooth evolution under the new regime this paper is concerned with the description of such impulsive smooth behavior on a typical interval we propose an algebraic framework modeled on the class of impulsive smooth distributions as defined by hautus both state space and polynomial representations are considered and we discuss transformations between the two forms

a numerical solution of the korteweg de vries equation by pseudospectral method using darvishi s preconditionings in this paper the korteweg de vries equation is solved by pseudospectral method in pseudospectral method we have to use matrix vector multiplication to compute derivative s of the unknown function to reduce roundoff error in this multiplication we use darvishi s preconditionings the numerical results have been compared with the exact solution they are in good agreement with each other because graphs of exact solution and numerical solutions are very similar

realization and partial fractions abstract we discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form it turns out that the relation is most naturally studied in the framework of partial fraction decompositions we develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may for instance correspond to stability and instability an isomorphism is obtained which connects the spaces used in the two methods and several identities relating to the mcmillan degree are derived in a direct way finally a new computational procedure is given to obtain the partial fraction decomposition of a rational matrix given in fractional form

fourier pseudospectral solution of the regularised long wave equation abstract an algorithm is presented for the fourier pseudospectral solution of the regularised long wave rlw equation the semi discrete equations satisfy the energy conservation condition of the rlw equation numerical results are presented to show that the scheme can resolve the fine structure of wave interaction problems evidence is presented to show that the popular second order leap frog time discretisation will produce high accuracy only if the time step is well below the linear stability limit by considering linear and nonlinear models a case is made for the use of higher order accuracy in time

a pointwise criterion for controller robustness abstract we present a pointwise criterion for controller robustness with respect to stability the term point here refers to complex frequency in the right half plane the proposed test is based on the concept of the minimal angle between subspaces determined by the plant and the compensator the test leads to separate balls of uncertainty at each frequency and may therefore help to reduce conservativeness in the analysis of robustness

some numerical aspects of computing inertial manifolds various aspects of the problem of computing inertial manifolds for dissipative partial differential equations are investigated in particular methods are proposed which generalize previous ones and improve upon some of their limitations the basic ideas and techniques are exemplified mainly for the kuramoto sivashinsky equation this equation is chosen because it is fairly simple but the dynamics are sufficiently complicated and it is a classical case for which a considerable amount of computational experience has been well documented

a fast algorithm for spectral differentiation a simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices the algorithm is used to multiply the chebyshev derivative matrix by a vector for the larger values of n the ratio of execution times approached the expected value of 2

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded