some upwinding techniques for finite element approximations of convection diffusion equations a uniform framework for the study of upwinding schemes is developed the standard finite element galerkin discretization is chosen as the reference discretization and differences between other discretization schemes and the reference are written as artificial diffusion terms these artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices three basis matrices are symmetric rank one diffusion operators associated with the edges of the triangle the fourth basis matrix is skew symmetric and is associated with a rotation by 2 while finite volume discretizations may be written as upwinded galerkin methods the converse does not appear to be true our approach is used to examine several upwinding schemes including the streamline diffusion method the box method the scharfetter gummel discretization and a divergence free scheme

measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

non linear dynamic analysis of anisotropic cylindrical shells containing a flowing fluid abstract an analytical model is presented to predict the influence of non linearities associated with fluid flow on the dynamic behaviour of a structure consisting of shells and a surrounding fluid medium the model requires the use of two linear operators to control the equilibrium of the shell and the velocity potential a linear boundary condition of impermeability and a non linear dynamic boundary condition the method is based on thin shell theory and the non rotational flow of non viscous fluids in combination with finite element analysis it is applicable to non uniform thin anisotropic cylinders subjected to different boundary conditions the displacement functions for the wall and liquid column are derived from sanders equations and from the velocity field associated with the column respectively the set of matrices describing their relative contributions to equilibrium is determined by exact analytical integration the coupled equations are solved for the non flow problem for cases of fluid flow certain analytical modifications are proposed to restore the situation to conventional modal analysis the non linear equations of motion are solved by the fourth order runge kutta numerical method the frequency variations are then determined with respect to the amplitude of the motion the trends of the non linearities are of a softening type

the sq universality and residual properties of relatively hyperbolic groups in this paper we study residual properties of relatively hyperbolic groups in particular we show that if a group g is non elementary and hyperbolic relative to a collection of proper subgroups then g is sq universal

sofic groups and diophantine approximation we prove the algebraic eigenvalue conjecture of j dodziuk p linnell v mathai t schick and s yates see 2 for sofic groups moreover we give restrictions on the spectral measure of elements in the integral group ring finally we define integer operators and prove a quantization of the operator norm below 2 to the knowledge of the author there is no group known that is not sofic

auto parametric resonance in cyclindrical shells using geometric averaging abstract a study is presented of internal auto parametric instabilities in the free non linear vibrations of a cylindrical shell focussed on two modes a concertina mode and a chequerboard mode whose non linear interaction breaks the in out symmetry of the linear vibration theory the two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards and at internal resonance significant energy transfer occurs between the modes a rayleigh ritz discretization of the von karman donnell equations leads to the hamiltonian and transformation into action angle co ordinates followed by averaging provides readily a geometric description of the modal interaction it was established that the interaction should be most pronounced when there are slightly less than 2 n square chequerboard panels circumferentially where n is the ratio of shell radius to thickness

nonlinear vibrations and imperfection sensitivity of a cylindrical shell containing axial fluid flow abstract the high imperfection sensitivity of cylindrical shells under static compressive axial loads is a well known phenomenon in structural stability on the other hand less is known of the influence of imperfections on the nonlinear vibrations of these shells under harmonic axial loads the aim of this work is to study the simultaneous influence of geometric imperfections and an axial fluid flow on the nonlinear vibrations and instabilities of simply supported circular cylindrical shells under axial load the fluid is assumed to be non viscous and incompressible and the flow to be isentropic and irrotational the behavior of the thin walled shell is modeled by donnell s nonlinear shallow shell equations it is subjected to a static uniform compressive axial pre load plus a harmonic axial load a low dimensional modal expansion which satisfies the relevant boundary and continuity conditions and takes into account all relevant nonlinear modal interactions observed in the past in the nonlinear vibrations of cylindrical shells with and without flow is used together with the galerkin method to derive a set of eight coupled nonlinear ordinary differential equations of motion which are in turn solved by the runge kutta method the shell is considered to be initially at rest in a position corresponding to a pre buckling configuration then a harmonic excitation is applied and conditions for parametric instability and dynamic snap through are sought the results clarify the marked influence of geometric imperfections and fluid flow on the dynamic stability boundaries bifurcations and basins of attraction

poms analysis of randomly vibrating systems obtained from karhunen loeve expansion abstract the karhunen loeve kl decomposition establishes that a 2d random field can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients the proper orthogonal decomposition pod method consists in detecting spatially coherent modes in the dynamics of a spatio temporally varying system by diagonalizing the spatial correlation function given by an averaging operator the kl expansion is applied here to the responses of randomly excited vibrating systems with a view to performing a pod in separated variables time and space form discrete and continuous mechanical systems are considered in this study as well as stationary and transient non stationary responses an averaging operator involving time and ensemble averages is proposed to draw up the pod in separated variables form from the associated kl expansion the result obtained using this approach agrees with the classical pod in the case of deterministic or ergodic random signals the associated proper orthogonal modes are interpreted in case of linear and nonlinear vibrating systems subjected to white noise excitation in terms of normal modes

dynamic instability and chaos of empty and fluid filled circular cylindrical shells under periodic axial loads abstract in the present paper the dynamic stability of circular cylindrical shells subjected to static and dynamic axial loads is investigated both donnell s nonlinear shallow shell and sanders koiter shell theories have been applied to model finite amplitude static and dynamic deformations results are compared in order to evaluate the accuracy of these theories in predicting instability onset and post critical nonlinear response the effect of a contained fluid on the stability and the post critical behaviour is analyzed in detail geometric imperfections are considered and their influence on the dynamic instability and post critical behaviour is investigated chaotic dynamics of pre compressed shells is investigated by means of nonlinear time series techniques extracting correlation dimension and lyapunov exponents

non linear vibrations of free edge thin spherical shells modal interaction rules and 1 1 2 internal resonance this paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement the analog for thin shallow shells of von karman s theory for large deflection of plates is used the validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution the specific case of a free edge is considered the governing partial differential equations are expanded onto the natural modes of vibration of the shell the problem is replaced by an infinite set of coupled second order differential equations with quadratic and cubic non linear terms analytical expressions of the non linear coefficients are derived and a number of them are found to vanish as a consequence of the symmetry of revolution of the structure then for all the possible internal resonances a number of rules are deduced thus predicting the activation of the energy exchanges between the involved modes finally a specific mode coupling due to a 1 1 2 internal resonance between two companion modes and an axisymmetric mode is studied 2004 elsevier ltd all rights reserved

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

methods for dimension reduction and their application in nonlinear dynamics abstract we compare linear and nonlinear galerkin methods in their efficiency to reduce infinite dimensional systems described by partial differential equations to low dimensional systems of ordinary differential equations both concerning the effort in their application and the accuracy of the resulting reduced system important questions like the choice of the form of the ansatz functions modes the choice of the number m of modes and finally the construction of the reduced system are addressed for the latter point both the linear or standard galerkin method making use of the karhunen loeve proper orthogonal decomposition ansatz functions and the nonlinear galerkin method using approximate inertial manifold theory are used in addition also the post processing galerkin method is compared with the other approaches

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

a uniformly convergent galerkin method on a shishkin mesh for a convection diffusion problem abstract a galerkin finite element method that uses piecewise bilinears on a simple piecewise equidistant mesh is applied to a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of ordern 1 ln nin a global energy norm and of ordern 1 2ln3 2 npointwise near the outflow boundary where the total number of mesh points iso n2

spectral multigrid methods for domain decomposition problems using patching techniques abstract we consider spectral domain decomposition problems that use patching techniques the resulting discrete systems are solved by the very efficient spectral multigrid methods for relaxation we employ a preconditioned richardson iteration effective preconditioning is achieved by one or more steps of a line relaxation method for the corresponding finite difference scheme numerical results are presented for rectangular and l shaped domains furthermore a cross point problem is considered

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

peripheral fillings of relatively hyperbolic groups in this paper a group theoretic version of dehn surgery is studied starting with an arbitrary relatively hyperbolic group g we define a peripheral filling procedure which produces quotients of g by imitating the effect of the dehn filling of a complete finite volume hyperbolic 3 manifold m on the fundamental group 1 m the main result of the paper is an algebraic counterpart of thurston s hyperbolic dehn surgery theorem we also show that peripheral subgroups of g almost have the congruence extension property and the group g is approximated in an algebraic sense by its quotients obtained by peripheral fillings

non linear dynamics and stability of circular cylindrical shells containing flowing fluid part i stability abstract the study presented is an investigation of the non linear dynamics and stability of simply supported circular cylindrical shells containing inviscid incompressible fluid flow non linearities due to large amplitude shell motion are considered by using the non linear donnell s shallow shell theory with account taken of the effect of viscous structural damping linear potential flow theory is applied to describe the fluid structure interaction the system is discretiszd by galerkin s method and is investigated by using a model involving seven degrees of freedom allowing for travelling wave response of the shell and shell axisymmetric contraction two different boundary conditions are applied to the fluid flow beyond the shell corresponding to i infinite baffles rigid extensions of the shell and ii connection with a flexible wall of infinite extent in the longitudinal direction permitting solution by separation of variables they give two different kinds of dynamical behaviour of the system as a consequence of the fact that axisymmetric contraction responsible for the softening non linear dynamical behaviour of shells is not allowed if the fluid flow beyond the shell is constrained by rigid baffles results show that the system loses stability by divergence

hidden symmetry concepts in the elastic buckling of axially loaded cylinders abstract the recognized harmonic buckling modes for the elastic axially loaded cylinder are by nature symmetric equal and opposite amplitudes giving equal energy levels on the other hand in combination they account for considerable asymmetry inwards deflection being fundamentally different from outwards this asymmetry is also apparent in the underlying differential equations and in the final large deflection yoshimura pattern taking the view that underlying symmetries are closely linked to the form of bifurcation experienced on buckling and hence to the gross instability of the phenomenon the paper thus explores a classic problem in a new light the wellknown donnell equations are first employed the analysis being neatly written in terms of the two variables radial displacement w and stress function an extension is then presented derived from the full strain displacement equations appropriate to genuinely large deflections this makes use of a suite of computer programs written for a small micro which handle the required manipulations of multiplication and integration of harmonic functions in algebraic rather than numerical fashion

chaotic energy exchange through auto parametric resonance in cylindrical shells abstract internal auto parametric instabilities in the free non linear vibrations of a cylindrical shell are studied numerically focusing on two modes a concertina mode and a chequerboard mode whose non linear interaction breaks the in out symmetry of the linear vibration theory the two mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards and at internal resonance significant energy transfer occurs between the modes this has regular and chaotic features here direct numerical integration is employed to examine chaotic motions using a set of 2 d poincare sections each valid for a fixed level of the hamiltonian h the instability under increasing h appears as a supercritical period doubling pitchfork bifurcation chaotic motions near a homoclinic separatrix appear immediately after the bifurcation giving an irregular exchange of energy this chaos occurs at arbitrarily low amplitude as perfect tuning is approached the instability manifests itself as repeating excursions around the separatrix and a number of practical predictions can be made these include the magnitude of the excursion the time taken to reach this magnitude and the degree of chaos and unpredictability in the outcome the effect of small damping is to pull the motion away from what was the chaotic separatrix giving a response that resembles for a while the lower energy quasi periodic orbits of the underlying hamiltonian system

a novel exponentially fitted triangular finite element method for an advection diffusion problem with boundary layers in this paper we develop an exponentially fitted finite element method for a singularly perturbed advection diffusion problem with a singular perturbation parameter this finite element method is based on a set of novel piecewise exponential basis functions constructed on unstructured triangular meshes the basis functions can not be expressed explicitly but the values of each of them and its associated flux at a point are determined by a set of two point boundary value problems which can be solved exactly a method for evaluating elements of the stiffness matrix is also proposed for the case that is small numerical results presented to validate the method show that the method is stable for a large range of it is also shown by the numerical results that the rate of convergence of the method in an energy norm is of orderh1 2when is small

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

low dimensional models of shell vibrations parametrically excited vibrations of cylinder shells vibrations of cylindrical shells parametrically excited by axial forcing are considered the governing system of two coupled non linear partial differential equations is discretized by using lagrange equations the computation is simplified significantly by the application of computer algebra and as a result low dimensional models of shell vibrations are readily obtained after applying numerical continuation techniques and ideas from dynamical systems theory complete bifurcation diagrams are constructed the principal aim is to investigate the interaction between different modes of shell vibration results for system models with two of the lowest modes are discussed

low dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition in formulating mathematical models for dynamical systems obtaining a high degree of qualitative correctness i e predictive capability may not be the only objective the model must be useful for its intended application and models of reduced complexity are attractive in many cases where time consuming numerical procedures are required this paper discusses the derivation of discrete low dimensional models for the nonlinear vibration analysis of thin cylindrical shells in order to understand the peculiarities inherent to this class of structural problems the nonlinear vibrations and dynamic stability of a circular cylindrical shell subjected to static and dynamic loads are analyzed this choice is based on the fact that cylindrical shells exhibit a highly nonlinear behavior under both static and dynamic loads geometric nonlinearities due to finite amplitude shell motions are considered by using donnell s nonlinear shallow shell theory a perturbation procedure validated in previous studies is used to derive a general expression for the nonlinear vibration modes and the discretized equations of motion are obtained by the galerkin method using modal expansions for the displacements that

chaotic vibrations of circular cylindrical shells galerkin versus reduced order models via the proper orthogonal decomposition method abstract the geometric nonlinear response of a water filled simply supported circular cylindrical shell to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency is investigated the response is investigated for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation bifurcation diagrams of poincare maps obtained from direct time integration and calculation of the lyapunov exponents and lyapunov dimension have been used to study the system by increasing the excitation amplitude the response undergoes i a period doubling bifurcation ii subharmonic response iii quasi periodic response and iv chaotic behaviour with up to 16 positive lyapunov exponents hyperchaos the model is based on donnell s nonlinear shallow shell theory and the reference solution is obtained by the galerkin method the proper orthogonal decomposition pod method is used to extract proper orthogonal modes that describe the system behaviour from time series response data these time series have been obtained via the conventional galerkin approach using normal modes as a projection basis with an accurate model involving 16 degrees of freedom dofs validated in previous studies the pod method in conjunction with the galerkin approach permits to build a lower dimensional model as compared to those obtainable via the conventional galerkin approach periodic and quasi periodic response around the fundamental resonance for fixed excitation amplitude can be very successfully simulated with a 3 dof reduced order model however in the case of large variation of the excitation even a 5 dof reduced order model is not fully accurate results show that the pod methodology is not as robust as the galerkin method

two level discretizations of nonlinear closure models for proper orthogonal decomposition proper orthogonal decomposition has been successfully used in the reduced order modeling of complex systems its original promise of computationally efficient yet accurate approximation of coherent structures in high reynolds number turbulent flows however still remains to be fulfilled to balance the low computational cost required by reduced order modeling and the complexity of the targeted flows appropriate closure modeling strategies need to be employed since modern closure models for turbulent flows are generally nonlinear their efficient numerical discretization within a proper orthogonal decomposition framework is challenging this paper proposes a two level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced order models the two level method computes the nonlinear terms of the reduced order model on a coarse mesh compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step the two level method attains the same level of accuracy while dramatically reducing the computational cost we numerically illustrate these improvements in the two level method by using it in three settings the one dimensional burgers equation with a small diffusion parameter n 10 3 the two dimensional flow past a cylinder at reynolds number re 200 and the three dimensional flow past a cylinder at reynolds number re 1000

stability and vibration of empty and fluid filled circular cylindrical shells under static and periodic axial loads abstract in the present study the dynamic stability of simply supported circular cylindrical shells subjected to dynamic axial loads is analysed geometric nonlinearities due to finite amplitude shell motion are considered by using the donnell s nonlinear shallow shell theory the effect of structural damping is taken into account a discretization method based on a series expansion involving a relatively large number of linear modes including axisymmetric and asymmetric modes and on the galerkin procedure is developed axisymmetric modes are included indeed they are essential in simulating the inward deflection of the mean oscillation with respect to the equilibrium position and in describing the axisymmetric deflection due to axial loads a finite length simply supported shell is considered the boundary conditions are satisfied including the contribution of external axial loads acting at the shell edges the effect of a contained liquid is investigated the linear dynamic stability and nonlinear response are analysed by using continuation techniques and direct simulations

normal generation and ell 2 betti numbers of groups the normal rank of a group is the minimal number of elements whose normal closure coincides with the group we study the relation between the normal rank of a group and its first ell 2 betti number and conjecture the inequality beta 1 2 g le mathrm nrk g 1 for torsion free groups the conjecture is proved for limits of left orderable amenable groups on the other hand for every n ge 2 and every varepsilon 0 we give an example of a simple group q with torsion such that beta 1 2 q ge n 1 varepsilon these groups also provide examples of simple groups of rank exactly n for every n ge 2 existence of such examples for n 3 was unknown until now

stability of constant amplitude motions in slow fluctuation approximation oscillatory motions at constant amplitude admit of an especially simple description of their stability in phase space orbital stability is equivalent to stability of the amplitudes alone regardless of phase behavior while liapunov stability can subsequently be inferred from the phases alone stability arguments simplify further in slow fluctuation approximation because of the availability of explicit quadratures for the amplitudes and phases depending ultimately on a single polynomial thus all orbital stability information about near resonant constant amplitude motions in conservative autonomous systems can be extracted solely from that one polynomial explicit analytic criteria for orbital stability are derived and auxiliary methods for the construction of stability charts are developed liapunov stability is shown to be a rare exception but liapunov instability is encountered in distinctly varying degrees a fairly wide class of motions in a fairly wide class of systems is shown to be liapunov u

small cancellations over relatively hyperbolic groups and embedding theorems we generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings this generalization is then used to prove various embedding theorems for countable groups for instance we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes in particular this gives the affirmative answer to the well known question of the existence of a finitely generated group g other than 2 such that all nontrivial elements of g are conjugate

theory and experiments for large amplitude vibrations of empty and fluid filled circular cylindrical shells with imperfections abstract the large amplitude response of perfect and imperfect simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated donnell s non linear shallow shell theory is used and the solution is obtained by the galerkin method several expansions involving 16 or more natural modes of the shell are used the boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied the effect of internal quiescent incompressible and inviscid fluid is investigated the non linear equations of motion are studied by using a code based on the arclength continuation method a series of accurate experiments on forced vibrations of an empty and water filled stainless steel shell have been performed several modes have been intensively investigated for different vibration amplitudes a closed loop control of the force excitation has been used the actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model several interesting non linear phenomena have been experimentally observed and numerically reproduced such as softening type non linearity different types of travelling wave response in the proximity of resonances interaction among modes with different numbers of circumferential waves and amplitude modulated response for all the modes investigated the theoretical and experimental results are in strong agreement

centroidal voronoi tessellation based reduced order modeling of complex systems noindent a reduced order modeling methodology based on centroidal voronoi tessellations cvts is introduced cvts are special voronoi tessellations for which the generators of the voronoi diagram are also the centers of mass means of the corresponding voronoi cells for discrete data sets cvts are closely related to the h means and k means clustering techniques a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced order bases then detailed descriptions of cvt based reduced order bases and how they can be constructed from snapshot sets and how they can be applied to the low cost simulation of complex systems are given subsequently some concrete incompressible flow examples are used to illustrate the construction and use of cvt based reduced order bases the cvt based reduced order modeling methodology is shown to be effective for these examples

real time modelling of diastolic filling of the heart using the proper orthogonal decomposition with interpolation abstract in this research a reduced order method rom called the proper orthogonal decomposition with interpolation podi is used to drastically reduce computation time of highly complex and non linear problems as encountered in simulating the heart the idea behind this method is to first construct a database of pre computed full scale solutions using the element free galerkin method efg and then project a selected subset of these solutions to a low dimensional space using the moving least square method an interpolation is carried out for the problem at hand before the resulting coefficients are projected back to the original high dimensional solution space computations are carried out on a bi ventricle model to investigate the performance and accuracy varying the material parameters and to determine the sensitivity with respect to the parametric values the podi calculations are completed within 1 25 s on a normal desktop machine with the relative l2 error norm not exceeding 2 5 10 3 hence it is demonstrated that real time modelling of the heart can be successfully carried out at acceptable error levels

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains part i convergence analysis the technique of separation of variables is used to derive explicit particular solutions for constant coefficient singularly perturbed partial differential equations pdes on a rectangular domain with dirichlet boundary conditions particular solutions and exact solutions in closed form are obtained an analysis of convergence for the series solutions is performed which is useful in numerical solution of singularly perturbed differential equations for moderately small values of e e g e 0 1 10 4 two computational models are designed deliberately model i with waterfalls solutions and model ii with wedding gauze solutions model ii is valid for very small e e g e 10 7 but model i for a moderately small e 0 1 10 4 the investigation contains two parts the first part reported in the present paper focuses on the convergence analysis and some preliminary numerical experiments for both of the models while the second part to be reported in a forthcoming paper will illustrate the solutions near the boundary layers

nonlinear normal modes for damped geometrically nonlinear systems application to reduced order modelling of harmonically forced structures in order to build efficient reduced order models roms for geometrically nonlinear vibrations of thin structures a normal form procedure is computed for a general class of nonlinear oscillators with quadratic and cubic nonlinearities the linear perturbation brought by considering a modal viscous damping term is especially addressed in the formulation a special attention is focused on how all the linear modal damping terms are gathered together in order to define a precise decay of energy onto the invariant manifolds also defined as nonlinear normal modes nnms then this time independent formulation is used to reduce the dynamics governing the oscillations of a structure excited by an external harmonic force the validity of the proposed roms is systematically discussed and compared with other available methods in particular it is shown that large values of the modal damping of the slave modes may change the type of nonlinearity hardening softening behaviour of the directly excited master mode two examples are used to illustrate the main features of the method a two degrees of freedom dof system allows presentation of the main results through a simple example then a water filled circular cylindrical shell with external resonant forcing is considered in order to show the ability of the method to substantially reduce the dynamics of a continuous structure 2006 elsevier ltd all rights reserved

on turing dynamical systems and the atiyah problem main theorems of the article concern the problem of atiyah on possible values of l 2 betti numbers it is shown that all non negative real numbers are l 2 betti numbers and that many for example all non negative algebraic real numbers are l 2 betti numbers of simply connected manifolds with respect to a free cocompact action also an explicit example is constructed which leads to a simply connected manifold with a transcendental l 2 betti number with respect to an action of the threefold direct product of the lamplighter group mathbb z 2 mathbb z wr mathbb z the main new idea is embedding turing machines into integral group rings the main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a comparison of shell theories for large amplitude vibrations of circular cylindrical shells lagrangian approach abstract large amplitude geometrically non linear vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated the lagrange equations of motion are obtained by an energy approach retaining damping through rayleigh s dissipation function four different non linear thin shell theories namely donnell s sanders koiter flugge lur e byrne and novozhilov s theories which neglect rotary inertia and shear deformation are used to calculate the elastic strain energy the formulation is also valid for orthotropic and symmetric cross ply laminated composite shells the large amplitude response of perfect and imperfect simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories numerical responses obtained by using these four non linear shell theories are also compared to results obtained by using the donnell s non linear shallow shell equation of motion a validation of calculations by comparison with experimental results is also performed both empty and fluid filled shells are investigated by using a potential fluid model the effects of radial pressure and axial load are also studied boundary conditions for simply supported shells are exactly satisfied different expansions involving from 14 to 48 generalized co ordinates associated with natural modes of simply supported shells are used the non linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis

non linear vibration analysis of fluid filled cylindrical shells abstract a theoretical analysis is presented for determining the elastic non linear vibrations of a prestressed thin walled cylindrical shell filled with an ideal fluid for the vibrations of the shell itself the dynamic version of the sanders non linear equations for the case of moderately small rotations is employed modal expansions are used for the displacements of the shell middle surface that are required to satisfy the classical simply supported boundary conditions and the circumferential periodicity condition the fluid is taken as non viscous and incompressible and the coupling between the deformable shell and this medium is taken into account the velocity potential is expanded in terms of harmonic functions which satisfy the laplace equation term by term the galerkin method is used to reduce the problem to a system of coupled algebraic non linear equations for the modal amplitudes solutions are presented to show the effects of fluid and shell parameters on the non linear vibrations of the shell

a nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection diffusion equation in this paper we formulate and analyze a discretization method for a 2d linear singularly perturbed convection diffusion problem with a singular perturbation parameter e the method is based on a nonconforming combination of the conventional galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method and on a mixture of triangular and rectangular elements it is shown that the method is stable with respect to a semi discrete energy norm and the approximation error in the semi discrete energy norm is bounded by ch ln e ln h with c independent of the mesh parameter h the diffusion coefficient e and the exact solution of the problem

accurate and approximate analytic solutions of singularly perturbed differential equations with two dimensional boundary layers in this paper we construct three new test problems called models a b and c whose solutions have two dimensional boundary layers approximate analytic solutions are found for these problems which converge rapidly as the number of terms in their expansion increases the approximations are valid for e 10 8 in practical computations surprisingly the algorithm for model a can be carried out even for e model c has a simple exact solution these three new accurate and approximate analytic solutions with two dimensional boundary layers may be more useful for testing numerical methods than those in z c li h y hu c h hsu s wang particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains i convergence analysis j comput appl math 166 2004 181 208 in the sense that the series solutions from the former converge much faster than those of the latter when e is small

the finite element method with weighted basis functions for singularly perturbed convection diffusion problems in this paper we present a finite element method for singularly perturbed convection diffusion problems in both one and two dimensions based on a set of weighted basis functions constructed on unstructured meshes in 2d for the one dimensional case both first and second order schemes are discussed a technique for approximating fluxes is proposed some theoretical results on uniform convergence are obtained for the two dimensional case a first order scheme is constructed for problems with two singular perturbation parameters a technique is also developed in approximating fluxes in 2d this technique is used to simplify the calculation of the integrals in the stiffness matrix arising from the scheme which will save computational costs the numerical results support the theoretical results and demonstrate that the method is stable for a wide range of singular perturbation parameters

an exponentially fitted finite volume method for the numerical solution of 2d unsteady incompressible flow problems abstract in this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the navier stokes equations describing 2d incompressible flows the method is based on an unstructured delaunay mesh and its dual dirichlet tessellation combined with a locally constant approximation to the flux this yields a piecewise exponential approximation to the exact solution numerical tests are presented for a linear advection diffusion problem with boundary layers the method is then applied to the driven cavity problem with reynolds numbers up to 10 4 the numerical results indicate that the method is robust for a wide range of values of the reynolds number in the case re 10 4 unsteady solutions are captured if the mesh is sufficiently fine

non linear dynamics and stability of circular cylindrical shells containing flowing fluid part ii large amplitude vibrations without flow abstract the non linear response of empty and fluid filled circular cylindrical shells to harmonic excitations is investigated both modal and point excitations have been considered the model is suitable to study simply supported shells with and without axial constraints donnell s non linear shallow shell theory is used the boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied the radial deflection of the shell is expanded by using a basis of seven linear modes the effect of internal quiescent incompressible and inviscid fluid is investigated the equations of motion obtained in part i of this study are studied by using a code based on the collocation method the validation of the present model is obtained by comparison with other authoritative results the effect of the number of axisymmetric modes used in the expansion on the response of the shell is investigated clarifying questions open for a long time the results show the occurrence of travelling wave response in the proximity of the resonance frequency the fundamental role of the first and third axisymmetric modes in the expansion of the radial deflection with one longitudinal half wave and limit cycle responses modes with two longitudinal half waves are also investigated

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

a novel nonconforming uniformly convergent finite element method in two dimensions we give an new analysis of a nonconforming galerkin finite element method for solving linear singularly perturbed two dimensional boundary value problems without turning points the method is shown to be convergent uniformly in the perturbation parameter of order h 1 2 in an energy norm the trial functions used are exponentials adapted to the differential operator

the elastic pendulum a nonlinear paradigm a pendulum with an elastic instead of an inextensible suspension is the simplest realization of an autonomous conservative oscillatory system of several degrees of freedom with nonlinear coupling it can also have an internal 1 2 resonance a fairly complete study of this system at and near resonance is here undertaken by means of the slow fluctuation approximation which consists in developing the x2y type interaction into a trigonometric polynomial and keeping only the term with the slowest frequency extensive computations showed that up to moderately large amplitudes the approximate solutions were virtually as accurate as numerical integrations of the exact equations of motion the slow fluctuation equations of motion can be completely integrated by quadratures explicit solutions for amplitudes and phases are given in terms of elliptic functions and can be linked to initial conditions there exist two branches of purely periodic harmonic constant amplitude motions which are orbitally stable b