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

on fuzzy linear representations of finite groups abstract a linear representation of a finite group g in a finite dimensional vector space v induces through zadeh s extension principle a function 9 from igto into igl v where gl v is the group of all linear automorphisms of v if w is a fuzzy subspace of v the group of all fuzzy linear automorphisms of w gl w is a subgroup of gl v w is said to be stable under the action of a fuzzy subgroup a of g if 9 a is a subset of gl w i e 9 a is zero at every f inside gl v and outside gl q if w is stable under the action of a then its support subspace is stable under the action of the support subgroup of a in the crisp sense finally we show that if there are two stable fuzzy subspaces one of them is contained in the other then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace

model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

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

invariant helical subspaces for the navier stokes equations three dimensional solutions with helical symmetry are shown to form an invariant subspace for the navier stokes equations uniqueness of weak helical solutions in the sense of leray is proved and these weak solutions are shown to be regular strong solutions existing for arbitrary time t the global universal attractor for the infinite dimensional dynamical system generated by the corresponding semi group of helical flows is shown to be compact and finite dimensional the hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class with the most detailed results obtained for rotating hagen poiseuille pipe flow in this case the dimension either hausdorff or fractal up to an absolute constant is bounded from above by frac operatorname re sqrt alpha 2 n 2 where is the axial wavenumber n is the azimuthal wavenumber and re is the reynolds number based on the radius of the pipe these upper bounds are independent of the rotation rate

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

attractor local dimensionality nonlinear energy transfers and finite time instabilities in unstable dynamical systems with applications to two dimensional fluid flows we examine the geometry of the finite dimensional attractor associated with fluid flows described by navier stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components modes of the flow specifically we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order reduction and describe the stochastic attractor in the reduced order phase space in terms of the associated probability measure we introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite time lyapunov exponents subsequently we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms in particular we illustrate the connection of the low dimensionality of the attractor with the circulation of energy i from the mean flow to the unstable modes due to their linearly unstable character ii from the unstable modes to the stable ones due to a nonlinear energy transfer mechanism and iii from the stable modes back to the mean due to the linearly stable character of these modes

inertial manifolds for the hyperviscous navier stokes equations abstract we prove the existence of inertial manifolds for the incompressible hyperviscous navier stokes equations on the two or three dimensional torus u t u u u p f t x r t d div u 0 where d 2 or 3 and 3 2 since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions we employ the spatial averaging method introduced by mallet paret and sell in 26

fourier spectral approximation to long time behavior of dissipative generalized kdv burgers equations in this paper we consider a generalized kdv burgers equation with periodic initial value condition firstly a fully discrete galerkin fourier spectral approximation scheme which is a linear one is constructed next the dynamical properties of the discrete system are analyzed for the autonomous case the existence and the convergence of the global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution the stability of the discrete scheme is also proved finally the long time stability and the convergence of the discrete scheme are proved for the nonautonomous case all results in this paper are obtained without any restriction on the time step size

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

noisy dynamic simulations in the presence of symmetry data alignment and model reduction abstract we process snapshots of trajectories of evolution equations with intrinsic symmetries and demonstrate the use of recently developed eigenvector based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise our illustrative examples include a one dimensional evolutionary partial differential the kuramoto sivashinsky equation with periodic boundary conditions as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear smoluchowski equation on the surface of a sphere this is a useful first step towards data mining the symmetry adjusted ensemble of snapshots in search of an accurate low dimensional parametrization and the associated reduction of the original dynamical system we also demonstrate a technique vector diffusion maps that combines in a single formulation the symmetry removal step and the dimensionality reduction step

approximation dynamics and the stability of invariant sets abstract we introduce the concept of a weakly normally hyperbolic set for a system of ordinary differential equations this concept includes the notion of a hyperbolic flow as well as that of a normally hyperbolic invariant manifold moreover it has the property that it is closed under finite set products consequently the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly normally hyperbolic sets our main objective is to show that under a small c 1 perturbation a weakly normally hyperbolic set k is preserved by a homeomorphism where the image k y is a compact invariant set with a related hyperbolic structure for the perturbed equation in addition the homeomorphism is close to the identity in c 0 1 and the perturbed dynamics on k y are close to the original dynamics on k

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

on the use of pod based roms to analyze bifurcations in some dissipative systems abstract this paper deals with the use of pod based reduced order models to construct bifurcation diagrams which requires calculating steady and time dependent attractors in complex bifurcation problems involving dissipative systems the method proposed in the paper relies on the observation that pod manifolds resulting from snapshots calculated in time dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters the reason for this property is explained for a general class of dissipative systems which includes many problems of scientific industrial interest the consequence is that appropriate pod manifolds can be calculated in a quite computationally efficient way the method is illustrated considering both a simple bifurcation problem for a fisher like equation and a fairly complex bifurcation problem for the complex ginzburg landau equation

on the reduction principle for differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a new type of differential equations with piecewise constant argument epcag more general than epca k l cooke j wiener retarded differential equations with piecewise constant delays j math anal appl 99 1984 265 297 j wiener generalized solutions of functional differential equations world scientific singapore 1993 the reduction principle v a pliss the reduction principle in the theory of the stability of motion izv akad nauk sssr ser mat 27 1964 1297 1324 in russian v a pliss integral sets of periodic systems of differential equations nauka moskow 1977 in russian is proved for epcag the structure of the set of solutions is specified we establish also the existence of global integral manifolds of quasilinear epcag in the so called critical case and investigate the stability of the zero solution

fuzzy ideals and fuzzy subalgebras of lie algebras abstract a fuzzy idea l of a lie algebra l factors it into a lie algebra l l called a fuzzy quotient lie algebra this fuzzy algebra lie algebra is always a fuzzy partition of l and a fuzzy partition of l will be a fuzzy quotient lie algebra if and only if its associated fuzzy similarity relation is compatible with the lie algebra operations relationship between the operations of the lie algebras l i t and zadeh s extension principle are studied we also study homomorphisms of lie algebras and the relationship between them and fuzzy ideals and fuzzy subalgebras of the domains and the co domains of these homomorphisms finally solvability of fuzzy ideals and studied in defining these solvable fuzzy ideals we use zadeh s extension principle

the adjoint representation of fuzzy lie algebras the commutator of a lie algebra l is extended by zadeh s extension principle to a product of fuzzy subsets a fuzzy subspace generated by the product of two fuzzy ideals is also a fuzzy ideal this notion of product of fuzzy ideals is used to define the descending central series of a fuzzy ideal nilpotency of fuzzy ideals is introduced using this descending central series fuzzification of the adjoint representation of lie algebras is introduced and some relations between this representation and nilpotent fuzzy ideals are proved finally the killing form in the fuzzy case is defined and some of its properties are studied

an efficient heuristic for scheduling in a flowshop to minimize total weighted flowtime of jobs the problem of scheduling in a flowshop is considered with the objective of minimizing the total weighted flowtime of jobs a heuristic algorithm is developed by the introduction of lower bounds on the completion times of jobs and the development of heuristic preference relations for the scheduling problem under study an improvement scheme is incorporated in the heuristic to enhance the quality of its solution the proposed heuristic with and without the improvement scheme and the existing heuristics are evaluated by a large number of randomly generated problems the results of an extensive computational investigation for various problem sizes are presented it has been observed that both versions of the proposed heuristic perform better than the existing heuristics in giving a superior solution quality and that the proposed heuristic without the improvement scheme yields a good solution by requiring a negligible cpu time in addition an experimental investigation is carried out to evaluate the effectiveness of the improvement scheme when implemented in the proposed heuristic and the existing heuristics as well as the effectiveness of a variant of the scheme the results are also discussed

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

heuristics for scheduling in flowshop with multiple objectives abstract most of the heuristics for flowshop scheduling aim at minimizing makespan however scheduling with multiple objectives such as that of minimizing makespan total flowtime and machine idletime is more effective in reducing the total scheduling cost in this article we first address the problem of scheduling to minimize makespan and total flowtime and propose a new heuristic algorithm a heuristic preference relation is developed as the basis for the heuristic so that only the potential job interchanges are checked for possible improvement with respect to these two objectives the proposed as well as the existing heuristics are evaluated in a large number of randomly generated large sized problems the proposed heuristic algorithm is then extended to cover the problem of scheduling to minimize makespan total flowtime and machine idletime the results of the experimental investigation of the evaluation of this heuristic algorithm and the existing heuristic in meeting these objectives are also reported it is found that both the proposed heuristics are more effective than the existing one for scheduling with multiple objectives

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

induced trajectories and approximate inertial manifolds for the ginzburg landau partial differential equations abstract this paper studies the long time behavior of solutions to the ginzburg landau partial differential equation we explicitly produce four approximate inertial manifolds and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time these neighborhoods localize the universal attractor we consider the space periodic case however with slight modifications the construction carries over to dirichlet and neumann boundary conditions

the two machine total completion time flow shop problem in this paper we study the np hard scheduling problem of minimizing total completion time in a two machine flow shop five known lower bounds are discussed and two new ones are presented a new dominance criterion is also proposed several versions of a branch and bound method are derived by applying both individually and combined these lower bounds a heuristic procedure is also presented that uses a constructive o n2 time method which computes a good starting solution together with a neighborhood search based on pairwise interchanges computational results show that the exact method can handle problems of up to 30 jobs in size within a reasonable amount of time and that the heuristic procedure has an average error of less than 0 5 from the optimal value and less than 2 7 from the lower bound

stability of galerkin and inertial algorithms with variable time step size this paper provides galerkin and inertial algorithms for solving a class of nonlinear evolution equations spatial discretization can be performed by either spectral or finite element methods time discretization is done by euler explicit or euler semi implicit difference schemes with variable time step size moreover the boundedness and stability of these algorithms are studied by comparison we find that the boundedness and stability of inertial algorithm are superior to the ones of galerkin algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi implicit scheme

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

formation of patterns in the rayleigh be nard convection abstract a model equation for a convection in a thin layer with nearly insulated boundaries is studied in the one dimensional case it is proved that the equation has periodic quasiperiodic and chaotic stationary solutions the stability of the periodic solutions with respect to long wave perturbations is investigated and the global dynamics of the flow is analyzed for a wide domain of parameters it is proved that almost all small initial data lead to a specific stationary periodic solution this solution corresponds to the wave number for which the zero solution is most unstable

nonlinear galerkin method in the finite difference case and wavelet like incremental unknowns the img algorithm inertial manifold multigrid algorithm which uses the first order incremental unknowns was introduced in 20 the img algorithm is aimed at numerically implementing inertial manifolds see e g 19 when finite difference discretizations are used for that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components first order incremental unknowns iu were proposed in 20 as a means to realize this decomposition our aim in the present article is to propose and study other forms of incremental unknowns in particular the wavelet like incremental unknowns wiu so called because of their oscillatory nature in this report we first extend the general convergence results in 20 by proving them under slightly weaker conditions we then present three sets of incremental unknowns i e the first order as in 20 the second order and wavelet like incremental unknowns we show that these incremental unknown can be used to construct convergent img algorithms special stress is put on the wavelet like incremental unknowns since this set of unknowns has thel 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms

global error analysis and inertial manifold reduction four types of global error for initial value problems are considered in a common framework they include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time to determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations we combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction these global error concepts are exercised to illustrate their utility on the lorenz equations and inertial manifold reductions of the kuramoto sivashinsky equation

correspondence theorem for fuzzy ideals abstract the ring r j of cosets of a fuzzy ideal j of a ring r is defined and shown to be isomorphic to a factor ring of r in a natural way if f r r is an epimorphism of rings a one to one order preserving correspondence is established between the fuzzy ideals of r and those of r which are constant on the kernel of f this generalizes the correspondence theorem of non fuzzy ideals

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

theory and methodology a bicriteria approach to the two machine flow shop scheduling problem in this paper we address the problem of minimizing makespan and sum of completion times simultaneously in a two machine flow shop environment we formulate the problem as a bicriteria scheduling problem and develop a branch and bound procedure that iteratively solves restricted single objective scheduling problems until the set of efficient solutions is completely enumerated we report computational results and explore certain properties of the set of efficient solutions we then discuss their implications for the decision maker

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

inertial manifolds for the 3d modified leray α model abstract in this paper we investigate the critical modified leray model in t 3 and prove the existence of an n dimensional inertial manifold for this problem

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

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

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

inertial manifolds for partial differential evolution equations under time discretization existence convergence and applications resume dans ce travail nous etudions l existence et la convergence de varietes inertielles pour la discretisation en temps d equations d evolution nous montrons que pour des pas de temps petits et sous une condition qui assure l existence de varietes inertielles exactes condition d ecart spectral le probleme discretise possede une variete inertielle de meme dimension nous montrons la convergence de la variete approchee vers la variete exacte dans un sens fort et avec estimation d erreur nos applications comprennent des equations non dissipatives elles ne sont pas limitees au cas purement parabolique c est ainsi que nous considerons des equations d amplitude du type ginzburg landau et des perturbations dissipatives des equations de korteweg de vries

fuzzy quotient groups abstract the operation of a group g induces through zadeh s extension principle a semigroup operation on ig we define a fuzzy quotient group of g to be a fuzzy partition of g that is a subsemigroup of ig having a group structure we show that fuzzy quotient groups of g are in one to one correspondence with normal fuzzy subgroups of g and that their associated fuzzy similarity relations are all two sided invariant we study relationships between the level structures of these structures we also give a necessary and sufficient condition for a group homomorphism to factor through fuzzy quotients

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

numerical analysis of a closed loop thermosyphon including the soret effect we analyse the motion of a fluid containing a soluble substance in the interior of a closed loop under the effects of natural convection and a given external heat flux numerical experiments are performed in order to describe the behaviour of solutions for different ranges of the relevant parameters

the two and m machine flowshop scheduling problems with bicriteria of makespan and mean flowtime abstract this paper considers the flowshop scheduling problem with the objective of minimizing the weighted sum of makespan and mean flowtime even though the two machine problem has been addressed and several heuristics have been established in the literature these heuristics have not been compared in this paper a comparison of these available heuristics in the literature is conducted moreover three new heuristic algorithms are proposed which can be utilized for both the two machine and m machine problems computational experiments indicate that the proposed heuristics are superior to all the existing heuristics in the literature including a genetic algorithm two dominance relations are also developed one for the two machine and the other for the three machine problem experimental results show that both relations are efficient

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

inertial manifolds and finite dimensional reduction for dissipative pdes this paper is devoted to the problem of finite dimensional reduction for parabolic partial differential equations we give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so called mane projection theorems the recent counter examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated as well as a discussion of the most important open problems are also included

on approximate inertial manifolds for two and three dimensional turbulent flows abstract in this paper we present some results concerning finite dimensional analytic manifolds that approximate the global attractor and the structure of the small eddies for the navier stokes equations under periodic boundary conditions we show that the smoother the external force the better the approximation

a bicriteria two machine permutation flowshop problem the problem attacked in this paper is the scheduling of n jobs on two machines which are assumed to be continuously available all jobs are available at the beginning of the scheduling period setup times are included in the processing times no preemption of jobs is allowed the objective is the minimization of a weighted combination of the average job flowtime and the schedule makespan three branch and bound bb and each one of these problems is solved for five different levels of the weighting factor an analysis into the resulting efficient points is also included an improved version of fsh1 called fsh2 is developed and both heuristics are subjected to extensive numerical experimentation

spectral barriers and inertial manifolds for time discretized dissipative equations the existence of inertial manifolds for dissipative nonlinear evolutionary equationsunder time discretizations of the semi implicit euler method is proved by using a generalized concept of spectral barrier for time discretized equations

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

approximation of attractors large eddy simulations and multiscale methods recent advances in the mathematical theory of the navier stokes equations have produced new insight in the mathematical theory of turbulence in particular the study of the attractor for the navier stokes equations produced the first connection between two approaches to turbulence that seemed far apart namely the conventional approach of kolmogorov and the dynamical systems theory approach similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows our aim in this article is to survey and describe some of the new results concerning the functional properties of the navier stokes equations and to discuss their relevance to turbulence

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

numerical convergence of the lyapunov spectrum computed using low mach number solvers abstract in the dynamical systems approach to describing turbulent or otherwise chaotic flows an important quantity is the lyapunov exponents and vectors that characterize the strange attractor of the flow in particular knowledge of the lyapunov exponents and vectors will help identify perturbations that the system is most sensitive to and quantify the dimension of the attractor however reliably computing these quantities requires robust numerical algorithms while several perturbation based techniques are available in literature their application to commonly used turbulent flow solvers as well the numerical convergence properties have not been studied in detail the goal of this work is to examine the convergence properties of the lyapunov exponents when computed using a low mach number solver for a series of progressively complex flow problems in particular a manufactured solutions approach is devised using the orr sommerfeld os perturbation theory to extract lyapunov exponents and vectors from os eigen solutions overall the spatial convergence rates of lyapunov exponents follow the truncation order for the discretization scheme however temporal convergence rates were found to be only a weak function of timestep used additionally for some configurations the convergence properties are found to depend on the lyapunov exponent itself these results indicate that convergence properties do not follow universal rates and require careful analysis for specific configurations considered

inertial manifolds associated to partly dissipative reaction diffusion systems abstract in a previous work we established the existence of finite dimensional attractors for partly dissipative reaction diffusion systems of two types systems with a polynomial growth nonlinearity and systems admitting an invariant region the aim of this article is to show that these problems possess inertial manifolds in space dimension n 2 these inertial manifolds contain the universal attractor attract exponentially all the solutions as time goes to infinity and due to the weak dissipation are here infinite dimensional our results are applied to several classical systems borrowed from mathematical biology physics and chemistry

convergence results for a class of spectrally hyperviscous models of 3 d turbulent flow abstract we consider the spectrally hyperviscous navier stokes equations shnse which add hyperviscosity to the nse but only to the higher frequencies past a cutoff wavenumber m 0 in guermond and prudhomme 2003 18 subsequence convergence of shnse galerkin solutions to dissipative solutions of the nse was achieved in a specific spectral vanishing viscosity setting our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of leray solutions in particular we obtain subsequence convergence of shnse strong solutions to leray solutions of the nse by fixing the hyperviscosity coefficient while the spectral hyperviscosity cutoff m 0 goes to infinity this formulation presents new technical challenges and we discuss how its motivation can be derived from computational experiments e g those in borue and orszag 1996 1998 3 4 we also obtain weak subsequence convergence to leray weak solutions under the general assumption that the hyperviscous coefficient goes to zero with no constraints imposed on the spectral cutoff in both of our main results the aubin compactness theorem provides the underlying framework for the convergence to leray solutions