permanence criteria for semi free profinite groups we introduce the condition of a profinite group being semi free which is more general than being free and more restrictive than being quasi free in particular every projective semi free profinite group is free we prove that the usual permanence properties of free groups carry over to semi free groups using this we conclude that if k is a separably closed field then many field extensions of k x y have free absolute galois groups

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

strong approximation for zariski dense subgroups of semi simple algebraic groups 1 1 theorem let k be an algebraically closed field of characteristic different from 2 and 3 and g an almost simple connected and simply connected algebraic group defined over k let f be a finitely generated zariski dense subgroup of g k and a the subring of k generated by the traces tr ad y y e p then there exist b e a a subgroup f of finite index in f and a structure gab of a group scheme over ab on g such that f c ga ab and f is dense in gala b a

using a dpg method to validate dma experimental calibration of viscoelastic materials abstract a discontinuous petrov galerkin dpg method is used to solve the time harmonic equations of linear viscoelasticity it is based on a broken primal variational formulation which is very similar to the classical primal variational formulation used in galerkin methods but has additional interface variables at the boundaries of the mesh elements both the classical and broken formulations are proved to be well posed in the infinite dimensional setting and the resulting discretization is proved to be stable a full h p convergence analysis is also included and the analysis is verified using computational simulations the method is particularly useful as it carries its own natural arbitrary p a posteriori error estimator which is fundamental for solving problems with localized solution features this proves to be useful when validating calibration models of dynamic mechanical analysis dma experiments indeed different dma experiments of epoxy and silicone resins were successfully validated to within 5 of the quantity of interest using the numerical method

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

characterization and computation of disturbance invariant sets for constrained switched linear systems with dwell time restriction this paper introduces the concept and characterization of disturbance dwell time invariance ddt invariance and constraint admissible ddt invariance caddt invariance for constrained systems with additive disturbance under dwell time switching using the characterization necessary and sufficient conditions for ddt invariance and algorithms for the computation of the minimal and maximal constraint admissible convex ddt invariant sets are provided numerical examples of such sets for several systems are also provided

robust stability and stabilization of linear switched systems with dwell time sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty a lyapunov function in quadratic form which is non increasing at the switching instants is assigned to each subsystem during the dwell time this function varies piecewise linearly in time after switching occurs it becomes time invariant afterwards this function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties the method proposed is then applied to stabilization via state feedback both for the nominal and the uncertain cases

non oscillatory and non diffusive solution of convection problems by the iteratively reweighted least squares finite element method in this paper we first introduce the least squares finite element method lsfem for two dimensional steady state pure convection problems with smooth solutions and compare the lsfem with other finite element methods we prove that the lsfem has the same stability estimate as the original equation i e the lsfem has better control of the streamline derivative than the streamline upwinding petrov galerkin method numerical convergence rates are given to show that the lsfem is almost optimal then we use this lsfem as a framework to develop an iteratively reweighted least squares finite element method irlsfem to obtain non oscillatory and non diffusive solutions for problems with contact discontinuities this new method produces a highly accurate numerical solution that has a sharp discontinuity in one element a number of examples solved by using triangular and bilinear elements are presented to show that the method can convect contact discontinuities without error

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

stabilization of switched continuous time systems with all modes unstable via dwell time switching stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems in this brief paper a sufficient condition ensuring the asymptotic stability of switched continuous time systems with all modes unstable is proposed the main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes then by using a discretized lyapunov function approach a computable sufficient condition for switched linear systems is proposed in the framework of dwell time it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability based on derived results an algorithm is proposed to compute the stability region of admissible dwell time a numerical example is proposed to illustrate our approach

on pseudo algebraically closed extensions of fields abstract the notion of pseudo algebraically closed pac extensions is a generalization of the classical notion of pac fields in this work we develop a basic machinery to study pac extensions this machinery is based on a generalization of embedding problems to field extensions the main goal is to prove that the galois closure of any proper separable algebraic pac extension is its separable closure as a result we get a classification of all finite pac extensions which in turn proves the bottom conjecture for finitely generated infinite fields the secondary goal of this work is to unify proofs of known results about pac extensions and to establish new basic properties of pac extensions e g transitiveness of pac extensions

absolute continuity of the periodic magnetic schrodinger operator we prove that the spectrum of the schrodinger operator with periodic electric and magnetic potentials is absolutely continuous

robust exponential stability and disturbance attenuation for discrete time switched systems under arbitrary switching in this note the global exponential stability of discrete time switched systems under arbitrary switching is investigated first for discrete time switched nonlinear systems the global exponential stability is found to be equivalent to the existence of an m step sequence with sufficient length and a family of lyapunov functions and then a stability criterion is proposed for the nominal linear case in the framework of quadratic lyapunov function in order to extend the stability criterion to handle uncertainties an equivalent condition which has a promising feature that is convex in system matrices is derived leading to a robust stability criterion for uncertain discrete time switched linear systems moreover also taking the advantage of the convex feature the disturbance attenuation performance in the sense of ell 2 gain is studied several numerical examples are provided to illustrate our approach

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

control of linear switched systems with receding horizon modal information we provide an exact solution to two performance problems one of disturbance attenuation and one of windowed variance minimization subject to exponential stability considered are switched systems whose parameters come from a finite set and switch according to a language such as that specified by an automaton the controllers are path dependent having finite memory of past plant parameters and finite foreknowledge of future parameters exact convex synthesis conditions for each performance problem are expressed in terms of nested linear matrix inequalities the resulting semidefinite programming problem may be solved offline to arrive at a suitable controller a notion of path by path performance is introduced for each performance problem leading to improved system performance non regular switching languages are considered and the results are extended to these languages two simple physically motivated examples are given to demonstrate the application of these results

hilbertian fields and galois representations we prove a new hilbertianity criterion for fields in towers whose steps are galois with galois group either abelian or a product of finite simple groups we then apply this criterion to fields arising from galois representations in particular we settle a conjecture of jarden on abelian varieties

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

nonconservative lifted convex conditions for stability of discrete time switched systems under minimum dwell time constraint in this note a novel conception called virtual clock which is defined by an artificial timer over a finite cycle is introduced for stability analysis of discrete time switched linear systems under minimum dwell time constraint two necessary and sufficient conditions associated with a virtual clock with a sufficient length are proposed to ensure the global uniform asymptotic stability of discrete time switched linear systems for the two nonconservative stability criteria the lifted version maintains the convexity in system matrices based on the lifted convex conditions the extensions to ell 2 gain computation and mathcal h infty control problems are presented in the sequel in particular a novel virtual clock dependent controller is designed which outperforms the traditional mode dependent and common gain controllers several numerical examples are provided to illustrate our theoretic results

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

stability and stabilization of discrete time switched systems this paper addresses two strategies for stabilization of discrete time linear switched systems the first one is of open loop nature trajectory independent and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition is expressed being non convex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

stability of discrete time switching systems with constrained switching sequences we introduce a novel framework for the stability analysis of discrete time linear switching systems with switching sequences constrained by an automaton the key element of the framework is the algebraic concept of multinorm which associates a different norm per node of the automaton and allows to exactly characterize stability building upon this tool we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius that is the exponential growth rate of a switching system with constrained switching sequences more precisely given a relative accuracy r 0 the algorithms compute an estimate of within the range 1 r these algorithms amount to solve a well defined convex optimization program with known time complexity and whose size depends on the desired relative accuracy r 0

derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems abstract the concept of the so called artificial or balancing diffusion used to stabilize the numerical solution of advective diffusive transport and fluid flow problems is revised in this paper it is shown that the standard forms of the balancing diffusion terms usually chosen in a heuristic manner can be naturally found by introducing higher order approximations in the derivation of the governing differential equations via standard conservation or equilibrium principles this allows us to reinterpret many stabilization algorithms and concepts used in every day practice by numerical analysts and also provides an expression for computing the stabilization parameter

stability analysis of switched linear systems defined by regular languages in this work we study the stability of an autonomous discrete time linear switched system whose switching sequences are generated by a muller automaton this system arises in various engineering problems such as distributed communication and automotive engine control the asymptotic stability of this system referred to as regular asymptotic stability ras generalizes two well known definitions of stability of autonomous discrete time linear switched systems namely absolute asymptotic stability aas and shuffle asymptotic stability sas we also extend these stability definitions to robust versions we show that absolute asymptotic stability robust absolute asymptotic stability and robust shuffle asymptotic stability are equivalent to exponential stability in addition by using the kronecker product we prove that a robust regular asymptotic stability problem is equivalent to the conjunction of several robust absolute asymptotic stability problems

computations of mode dependent dwell times for discrete time switching system given a system that switches among n linear subsystems this paper shows an approach that computes n dwell times one for each of the subsystems such that the overall system is stable under switching signals that respect the dwell times the dwell times are obtained progressively starting from the groups of all pairwise switching systems and increasing the size of the group by one for each step in each progressive step a bisection search algorithm is used to obtain the mode dependent dwell times for that step when the final step is reached the n mode dwell times are obtained these dwell times are smaller in terms of their sum than an existing approach in recent literature for all the examples considered in this paper

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

stability stabilization and l 2 gain analysis of periodic piecewise linear systems in this paper the stability stabilization and l 2 gain problems are investigated for periodic piecewise linear systems in which not all subsystems are hurwitz first some sufficient and necessary conditions for the exponential stability are established by employing a discontinuous lyapunov function with time varying lyapunov matrix stabilization and l 2 gain conditions of periodic piecewise linear systems are proposed by allowing the corresponding lyapunov function to be possibly non monotonically decreasing over a period a state feedback periodic piecewise controller is developed to stabilize the system and the corresponding algorithm is proposed to compute the controller gain the l 2 gain criteria with continuous time varying lyapunov matrix and piecewise constant lyapunov matrices are studied as well numerical examples are given to show the validity of the proposed techniques

the discontinuous petrov galerkin method for elliptic problems abstract we propose a novel discontinuous mixed finite element formulation for the solution of second order elliptic problems fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions the discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous galerkin methods static condensation is performed at the element level leading to a solution procedure based on the sole interface unknowns the resulting family of discontinuous dual primal mixed dpm finite element methods is presented in the one and two dimensional cases in the one dimensional case we show the equivalence of the method with implicit runge kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis numerical experiments in one and two dimensions demonstrate the order accuracy of the new method confirming the results of the analysis

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

multiscale stabilization for convection dominated diffusion in heterogeneous media abstract we develop a petrov galerkin stabilization method for multiscale convection diffusion transport systems existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion which may not be sufficient to stabilize multiscale systems we seek a local reduced order model for this kind of multiscale transport problems and thus develop a systematic approach for finding reduced order approximations of the solution we start from a petrov galerkin framework using optimal weighting functions we introduce an auxiliary variable to a mixed formulation of the problem the auxiliary variable stands for the optimal weighting function the problem reduces to finding a test space a dimensionally reduced space for this auxiliary variable which guarantees that the error in the primal variable representing the solution is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution to find the test space we reformulate some recent mixed generalized multiscale finite element methods we introduce snapshots and local spectral problems that appropriately define local weight and trial spaces in particular we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable the resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system we discuss the stability and its relation to the approximation property of the test space we design online basis functions which accelerate convergence in the test space and consequently improve stability we present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the peclet number

diamond theorem for a finitely generated free profinite group we extend haran s diamond theorem to closed subgroups of a finitely generated free profinite group this gives an affirmative answer to problem 25 4 9 of fried and jarden in field arithmetic 2nd edn springer berlin heidelberg new york 2005

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

analysis of the discontinuous petrov galerkin method with optimal test functions for the reissner mindlin plate bending model we analyze the discontinuous petrov galerkin dpg method with optimal test functions when applied to solve the reissner mindlin model of plate bending we prove that the hybrid variational formulation underlying the dpg method is well posed stable with a thickness dependent constant in a norm encompassing the l 2 norms of the bending moment the shear force the transverse deflection and the rotation vector we then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p we show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p 3 we prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non affine mesh sequences

post classification version of jordan s theorem on finite linear groups abstract using classification of finite simple groups i show that a finite subgroup g of gln c where c the complex numbers contains a commutative normal subgroup m of index at most n 1 nalogn b moreover if g is primitive and does not contain normal subgroups that are direct products of large alternating groups then the factor n 1 can be dropped i further show that similar statements hold also in characteristics p 2 if one takes m to be an extension of a group of lie type of characteristic p by a solvable group that has a normal p subgroup with commutative p quotient these results improve the celebrated theorems of jordan and of brauer and feit

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

parameter memorized lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper a novel parameter memorized lyapunov function is proposed for stability analysis of discrete time linear systems with time varying parametric uncertainties the parameter memorized lyapunov function depends on the uncertain parameters with a memory of a certain interval it is shown that the previous parameter dependent lyapunov function is a special memoryless case of the parameter memorized lyapunov function and as a result the parameter memorized lyapunov function approach leads to less conservative results furthermore if the length of the interval for parameter memory is sufficiently long a nonconservative stability analysis result can be achieved numerical examples are provided to evaluate the obtained theoretical results

an adaptive characteristic petrov galerkin finite element method for convection dominated linear and nonlinear parabolic problems in one space variable a new adaptive finite element method for convection dominated problems is presented a special feature of the method is that it is based on a petrov galerkin scheme for spatial approximation at a typical time step which employs test functions chosen so that the approximate solution coincides with the exact solution at the nodes of finite element grid this procedure makes possible the derivation of truly local a posteriori error estimates and a very effective solver numerical examples are discussed which illustrate the efficiency and effectively of the method

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

diagonal actions in positive characteristic we prove positive characteristic analogues of certain measure rigidity theorems in characteristic 0 more specifically we give a classification result for positive entropy measures on quotients of sld and a classification of joinings for higher rank actions on simply connected absolutely almost simple groups

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

word maps and waring type problems waring s problem asks whether every natural number is a sum of g k fcth powers where g is a suitable function this was solved affirmatively by hubert in 1909 optimizing g k has been a central problem in additive number theory ever since see na for more detail and background recently there has been considerable interest in group theoretic analogues of this phenomenon where the aim is to present group elements as short products of certain special elements these special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group the motivation for this is sometimes topological let g be a topologically finitely generated pro p group using expressions of elements of the commutator subgroup g1 as bounded products of commutators serre showed that g is closed and deduced that every finite index subgroup of g is open a recent deep result of nikolov and segal nsi ns2 shows that this holds for every finitely generated profinite group again the core of the proof is presenting group elements as short products of values of certain words

brief paper modal and transition dwell time computation in switching systems a set theoretic approach we consider a plant the dynamics of which switch among a family of systems each of these systems has a single stable equilibrium point we assume that a constraint region for the state is assigned and we consider the problem of finding suitable limitations on the commutation speed in order to avoid constraints violations even in the absence of state measurements we introduce the concepts of modal and transition dwell times which lead to the definition of dwell time vector and dwell time graph represented by a proper matrix respectively the former imposes a minimum permanence time on a discrete mode before commuting the latter imposes the minimum permanence time on the current mode before switching to a specific new one both dwell time vector and dwell time graph can be computed via set theoretic techniques when the systems share a single equilibrium state stability can be assured as a special case finally under the assumption of affine dynamics non conservative values are achieved

hp fem for reaction diffusion equations i robust exponential convergence a singularly perturbed reaction diffusion equation in two dimensions is considered we assume analyticity of the input data i e the boundary of the domain is an analytic curve and the right hand side is analytic we show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width o p eps is inserted near the domain boundary that is the rate of convergence is o mbox rm exp b p and independent of the perturbation parameter eps additionally we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence in particular the spectral element method based on the use of a gauss lobatto quadrature rule with p 1 x p 1 points yields robust exponential convergence

necessary and sufficient condition for stability of switched uncertain linear systems under dwell time constraint in this technical note a necessary and sufficient stability criterion for switched linear systems under dwell time constraint is proposed by employing a class of time scheduled homogeneous polynomial lyapunov functions with a sufficiently large degree the key feature of this nonconservative condition lies in its convexity in the system matrices which explicitly facilitates its further extension to uncertain systems then in order to obtain numerically testable condition a family of lmi conditions are presented with the aid of the idea of dividing the dwell time interval into a finite number of segments it is proved that the non conservativeness can be maintained with a sufficiently large interval dividing parameter in the end the result is straightforwardly extended to the uncertain case in virtue of the convexity in the system matrices numerical examples are presented to illustrate our findings

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients abstract we present a new stabilization technique for multiscale convection diffusion problems stabilization for these problems has been a challenging task especially for the case with high peclet numbers our method is based on a constraint energy minimization idea and the discontinuous petrov galerkin formulation in particular the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions and are related to the trial space the resulting test functions have a localization property and can therefore be computed locally we will prove the stability and present several numerical results our numerical results confirm that our test space gives a good stability in the sense that the solution error is close to the best approximation error

modeling subgrid viscosity for advection diffusion problems we analyse the effect of the subgrid viscosity on a finite element discretisation with piecewise linear elements of a linear advection diffusion scalar equation we point out the importance of a proper tune up of the viscosity coefficient and we propose a heuristic method for obtaining reasonable values for it the extension to more general problems is then hinted in the last section

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

affine characterizations of minimal and mode dependent dwell times for uncertain linear switched systems an alternative approach for minimum and mode dependent dwell time characterization for switched systems is derived while minimum dwell time results require the subsystems to be asymptotically stable mode dependent dwell time results can consider unstable subsystems and dwell times within a possibly unbounded range of values the proposed approach is related to lyapunov looped functionals a new type of functionals leading to stability conditions affine in the system matrices unlike standard results for minimum dwell time these conditions are expressed as infinite dimensional lmis which can be solved using recent polynomial optimization techniques such as sum of squares the specific structure of the conditions is finally utilized in order to derive dwell time stability results for uncertain switched systems several examples illustrate the efficiency of the approach

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

convex lifted conditions for robust l 2 stability analysis and l 2 stabilization of linear discrete time switched systems with minimum dwell time constraint stability analysis of discrete time switched systems under minimum dwell time is studied using a new type of lmi conditions these conditions are convex in the matrices of the system and shown to be equivalent to the nonconvex conditions proposed in geromel and colaneri 2006b the convexification of the conditions is performed by a lifting process which introduces a moderate number of additional decision variables the convexity of the conditions can be exploited to extend the results to uncertain systems control design and 2 gain computation without introducing additional conservatism several examples are presented to show the effectiveness of the approach

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100