adaptive finite element methods for hyperbolic systems with application to transient acoustics the solution of the hyperbolic systems of equations governing transient acoustics by adaptive finite elements and various finite difference time discretization schemes is addressed emphasis is placed on the use of a class of implicit runge kutta methods for temporal approximations

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

paths and partitions combinatorial descriptions of the parafermionic states the zk parafermionic conformal field theories despite the relative complexity of their modes algebra offer the simplest context for the study of the bases of states and their different combinatorial representations three bases are known the classic one is given by strings of the fundamental parafermionic operators whose sequences of modes are in correspondence with restricted partitions with parts at distance k 1 differing at least by 2 another basis is expressed in terms of the ordered modes of the k 1 different parafermionic fields which are in correspondence with the so called multiple partitions both types of partitions have a natural bressoud path representation finally a third basis formulated in terms of different paths is inherited from the solution of the restricted solid on solid model of andrews baxter forrester the aim of this work is to review in a unified and pedagogical exposition these four different combinatorial representations of the states of the zk parafermionic models

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

modeling of electromagnetic fields in rectangular waveguides by coupled finite infinite elements abstract the paper describes the 3d infinite element for modeling of stationary harmonic electromagnetic fields in waveguides the proposed approximation is a straightforward modification of the analogous approach developed for analysis of scattering problems in unbounded 2d and 3d domains comp meth appl mech engrg 188 2000 625 int j num meth engrg 57 2003 899 exponential shape functions in the longitudinal direction are used similarly as in comp meth appl mech engrg 140 1997 221 however arbitrary order of approximation in the tangential direction may be selected due to compatibility with the hp adaptive edge fe element discretization for maxwell s equations in bounded domains reported in comp meth appl mech engrg 152 1998 103 int j num meth engrg 53 2002 147 comp meth appl mech engrg 169 1999 331

kac wakimoto characters and universal mock theta functions in recent work bringmann and ono answer a question of kac and show that character formulas for sl r 1 1 modules due to kac and wakimoto are holomorphic parts of nonholomorphic modular functions here we confirm a speculation of ono that these characters are up to a simple q series the universal mock theta functions g 2 w q and g 3 w q of gordon and mcintosh using recent work of bringmann ono kang zwegers and gordon mcintosh we show that g 2 w q and g 3 w q are up to classical theta functions and products the characters of kac and wakimoto as a consequence we include a dictionary that gives a character formula for every classical mock theta function of ramanujan as well as subsequent natural generalizations

a posteriori error estimation of h p finite element approximations of frictional contact problems abstract dynamic and static frictional contact problems are described using the normal compliance law on the contact boundary dynamic problems are recast into quasistatic problems by time discretization an a posteriori error estimator is developed for a nonlinear elliptic equation of corresponding static or quasistatic problems the a posteriori error estimator is applied to a frictionless case and extended to frictional contact problems also an adaptive strategy is introduced and h p finite element meshes are obtained through a procedure based on a priori and a posteriori error estimations numerical examples are given to support the theoretical results

prophlex an hp adaptive finite element kernel for solving coupled systems of partial differential equations in computational mechanics this paper presents an overview of the basic design and architecture of the prophlex hp adaptive finite element kernel prophlex was designed to be a commercial robust implementation of hp adaptivity driven by residual error estimation with the primary goal of being physics independent and computationally efficient on a wide array of computer hardware platforms prophlex can solve virtually any class of engineering problems which be may be mathematically formulated as a system of linear or nonlinear second order partial differential equations and associated boundary conditions it has been applied to compressible and incompressible fluid dynamics linear and nonlinear solid mechanics heat flow problems as well as semiconductor device simulation examples of prophlex customization for linear and nonlinear solid mechanics are presented

the solution of a launch vehicle trajectory problem by an adaptive finite element method we describe a second order discontinuous galerkin finite element method for the solution of an optimal control problem for determining the trajectory of a launch vehicle we derive an a posteriori error estimate that is subsequently implemented as the basis for adaptive error control we demonstrate that the computational error estimate is reliable and accurate while the adaptive error control provides a significant gain in efficiency over uniform discretizations

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

schwarz methods with local refinement for the p version finite element method in some applications the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain in this paper local refinement is introduced for an overlapping additive schwarz algorithm for the p version finite element method both uniform and variable degree refinements are considered the resulting algorithm is highly parallel and scalable

a straightforward structure to construct shape functions for variable p order meshes this paper describes a method for specifying variable p order meshes based on the topological hierarchy of mesh entities defining the finite elements a decomposition of p version mesh entity shape functions into a product of a function defined over the mesh entity and a blending function defined over the elements that the mesh entity bounds is presented the decomposed shape functions allow the effective construction of the shape functions and derivatives needed to construct element matrices for variable p enrichment

on an adaptive monotonic convection diffusion flux discretization scheme abstract the present paper concerns numerical investigation of a two dimensional steady convection diffusion scalar equation specific attention is given to resolving spurious oscillations in a confined region of high gradient in smooth regions a high order accurate solution is desired while in regions containing a sharp gradient the strategy of applying a monotonic capturing scheme is preferred in this paper we model fluxes by means of a finite element model which has spaces spanned by legendre polynomials the propensity to yield an irreducible diagonal dominant stiffness matrix is an attribute of this finite element flux discretization scheme fundamental analyses are extensively conducted in this paper in two areas first by conducting a modified equation analysis we are led to confirm the consistency of the scheme both the stability and monotonicity of the solutions are also addressed a guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the m matrix this monotonicity study provides us with greater numerical insight into the importance of the chosen peclet number beyond its critical value oscillatory solutions are present in the flow this monotonic region is however fairly restricted it is this shortcoming which limits the finite element practioner s choices for fairly small grid size the scope of application using the proposed upwind scheme is thus rather small because monotonic solutions are prohibitively expensive to compute circumvention of such deficiency is accomplished by making modifications to the structured grid several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting

recent progress on application of hp adaptive be fe methods to elastic scattering summary several new issues related to the projcct described in references 1 3 are reported the presented problems include i sensitivity of p convergence results to geometric parametrizations and an idea of a new mesh generation for lip approximations 2 a posteriori error estimation for the coupled problem 3 a better understanding of the meaning of the asymptotic convergence a few typical numerical examples conclude the presentation

a priori error estimations of hp finite element approximations for hierarchical models of plate and shell like structures a priori error estimates of hp finite element approximations for hierarchical models of the thin elastic plate and shell like structures are derived the error in the approximations of a hierarchical model consists of the modelling error due to assumptions on displacement or stress fields and the finite element approximation error of discretizations of such a model also non uniformity with respect to the thickness owing to numerical locking is included in the present error estimates numerical results are given supporting the theoretical results

a framework of rogers ramanujan identities and their arithmetic properties the two rogers ramanujan q series where 0 1 play many roles in mathematics and physics by the rogers ramanujan identities they are essentially modular functions their quotient the rogers ramanujan continued fraction has the special property that its singular values are algebraic integral units we find a framework which extends the rogers ramanujan identities to doubly infinite families of q series identities if a 1 2 and m n 1 then we have infinite product modular function where the p x x q are hall littlewood polynomials these q series are specialized characters of affine kac moody algebras generalizing the rogers ramanujan continued fraction we prove in the case of a that the relevant q series quotients are integral units

multi parameter local semi groups of hermetian operators abstract it is proven that if a weakly continuous multi parameter family of hermetian operators in a hilbert space whose domains are not necessarily dense but whose union is dense and which has the semi group property for the parameters belonging to an open convex set in euclidean n space with 0 a limit point then the operators have self adjoint extensions which have a joint spectral representation

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

wamr an adaptive wavelet method for the simulation of compressible reacting flow part i accuracy and efficiency of algorithm abstract the wavelet adaptive multiresolution representation wamr method provides a robust method for controlling spatial grid adaptation fine grid spacing in regions where a solution varies greatly i e near steep gradients or near singularities and a much coarser grid where the solution varies slowly subsequently a wide range of spatial scales often demanded in challenging continuum physics problems can be efficiently captured furthermore the wavelet transform provides a direct measure of local error at each collocation point effectively producing automatically verified solutions the method is applied to the solution of unsteady compressible reactive flow equations and includes detailed diffusive transport and chemical kinetics models accuracy and performance of the method are examined on several test problems the sparse grids produced by the wamr method exhibit an impressive compression of the solution reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution

hierarchical finite element analysis of the vibration of membranes abstract a hierachical finite element for the vibration of membranes is presented the element has the ability to be joined to neighbouring elements and the numbers of hierarchical terms are allowed to vary in both directions of the element co ordinate axes the element transverse displacement is described by four linear shape functions plus a variable number of hierarchical functions which are forms of legendre orthogonal polynomials the four nodal displacements and the amplitudes of the hierarchical functions on the edges and in the interior of the element are used as generalized co ordinates inter element compatibility is achieved by matching the generalized co ordinates at the nodes and edges shared by elements results are obtained for simply supported square and l shaped membranes comparisons are made with exact solutions for the square membrane and with highly accurate approximate and linear finite element solutions for the l shaped membrane the results for the square membrane confirm that the solutions always converge from above to the exact values as the numbers of hierarchical terms are increased and highly accurate answers are obtained depite the use of a very few hierarchical terms the results of the l shaped membrane show that the hierarchical finite element solutions are largely more accurate than the linear finite element solutions despite the use of fewer system degrees of freedom

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

essential boundary conditions and multi point constraints in finite element analysis the application of boundary conditions and other constraints to the stiffness matrix and load vector is an integral part of any finite element code this process is usually trivial but can present difficulties to the point where some codes disallow certain combinations of commonly occurring boundary conditions a general approach to this problem is presented and applied to representative examples necessary and sufficient conditions are obtained under which constraints can be applied sequentially bounds are obtained for the condition number of the reduced system showing it is essentially the same as for the original problem

stability and convergence of nonconforming hp finite element methods the stability and convergence of nonconforming hp finite element methods in par ticular the mortar finite element method and its variants are established based on a new stability measure for these methods using a generalized eigenvalue analysis estimates for this measure are computed numerically our numerical results demonstrate that these nonconforming methods prove to be good candidates for hp implementation and also behave ss well as conforming finite element methods the discussion here ls primarily in two dimensions but some extensions to three dimensions

from the euclidean group to the poincare group via osterwalder schrader positivity given a continuous representation of the euclidean group inn 1 dimensions together with a covariant system of subspaces which satisfies osterwalder schrader positivity we construct a continuous unitary representation of the orthochronous poincare group inn 1 dimensions satisfying the spectral condition a similar result holds for the covering groups of the euclidean and poincare group

three dimensional free vibration analysis of plates using the h p version of the finite element method the h p version of the finite element method based on a pentahedral p element is applied to three dimensional free vibrations of plates the element s new hierarchical shape functions are expressed in terms of the shifted legendre orthogonal polynomials the accuracy of the solution can be improved by simultaneously refining the mesh and increasing the polynomial order the method is capable of providing a full frequency spectrum including both the in plane and the out of plane modes results of frequency calculations are first found for a skew plate on a soft edge support and a cantilevered right isosceles triangular plate and comparisons are made with other methods results are also found for a square plate on a hard edge support and comparisons are made with two dimensional plate theories furthermore highly accurate values for a free hexagonal plate are presented for the first time

adaptive hpq finite element methods of hierarchical models for plate and shell like structures abstract a four step adaptive strategy for hpq finite element approximations of hierarchical models of the thin plate and shell like structures is derived based on a priori and a posteriori error estimations this strategy controls three parameters the model level q the mesh size h and the approximation order p so as to produce quasi optimal hierarchical models and finite element meshes numerical results are given supporting the theoretical results

estimation of computational homogenization error by explicit residual method in this paper a new method is introduced for the estimation of modeling error resulting from homogenization of elastic heterogeneous bodies the approach is similar to the well known explicit residual approximation error estimation it is proved that besides the residuum of the equilibrium equation and interelement traction jump also a difference of stress divergences as well as traction jump along the material interfaces contribute to the modeling error estimate moreover explicit specification and numerical evaluation of stability constants provide reasonable effectivity index of this error indicator selected numerical examples illustrate the promise of this approach therefore the proposed methodology is a computationally inexpensive option for the other methods of modeling error assessment

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

on virtual representations of symmetric spaces and their analytic continuation in this paper we study a problem in group representation theory motivated by relativistic quantum field theory one of the most powerful approaches to constructing models of relativistic quantum fields relies on euclidean field theory a description of quantum field theory in which time is purely imaginary the basic objects of euclidean field theory the euclidean green s or schwinger functions can often be expressed in terms of functional integrals the construction of quantum field models is thereby reduced to quadratures a well known simple example of this method is the reformulation of quantum mechanisms by means of wiener integrals the fundamental solution of the schrodinger equation analytically continued to imaginary time is given by the well known feynman kac formula see e g 15 19 in this approach one faces the problem of analytic continuation back to real time while this problem is elementary in non relativistic quantum mechanics it is rather intricate in relativistic quantum field theory for some class of relativistic quantum field theories a fairly general solution to this problem was given in papers by osterwalder and

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

adaptivity techniques for compressible inviscid flows abstract to enhance the resolution of numerically simulated flowfields grid adaptivity is vitally important further to afford complete flexibility in the spatial discretisation of the flow domain both structured unstructured and combinations of grid types are likely to be required hence the development of grid adaptivity techniques for general grids is important this paper discusses adaptivity achieved through point enrichment point derefinement and point movement applied on general grids a flexible data structure is described together with suitable strategies for adaptively refining a computational grid applications are presented of adaptive grid refinement for two dimensional compressible flow simulation

lie algebras of unbounded derivations abstract in this paper some new results on analytic domination of operators and on integrability of lie algebras of operators are proved and then our methods are applied to the study of lie algebras of unbounded derivations in c algebras

a posteriori error estimators for second order elliptic systems part 2 an optimal order process for calculating self equilibrating fluxes abstract we present full details of an algorithm which can be used to obtain self equilibrating fluxes for arbitrary h p finite element approximations all of the calculations are performed on a local scale yielding an optimal order process o n where n is the number of elements in the mesh owing to the local nature of the process the algorithm is perfectly suited to parallel implementation the algorithm can be applied in both two and three dimensions for unstructured k irregular meshes and elements of non uniform degree finally it is shown that the implementation of the flux equilibration algorithm can take advantage of calculations already performed in obtaining the finite element approximation itself

thermal quantum fields with spatially cutoff interactions in 1 1 space time dimensions we construct interacting quantum fields in 1 1 space time dimensions representing charged or neutral scalar bosons at positive temperature and zero chemical potential our work is based on prior work by klein and landau and hoegh krohn generalized path space methods are used to add a spatially cut off interaction to the free system which is described in the araki woods representation it is shown that the interacting kms state is normal w r t the araki woods representation the observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting liouvillean is described in terms of the free liouvillean and the interaction

a posteriori error estimators for second order elliptic systems part 1 theoretical foundations and a posteriori error analysis abstract this is the first in a series of two papers dealing with a posteriori error estimation for h p finite element approximation of second order elliptic systems in this paper we shall present the fundamental ideas behind the error estimator and the theoretical foundations for the method the second paper deals with the algorithmical details

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

complete monotonicity for inverse powers of some combinatorially defined polynomials we prove the complete monotonicity on 0 infty n for suitable inverse powers of the spanning tree polynomials of graphs and more generally of the basis generating polynomials of certain classes of matroids this generalizes a result of szeg and answers among other things a long standing question of lewy and askey concerning the positivity of taylor coefficients for certain rational functions our proofs are based on two ab initio methods for proving that p beta is completely monotone on a convex cone c the determinantal method and the quadratic form method these methods are closely connected with harmonic analysis on euclidean jordan algebras or equivalently on symmetric cones we furthermore have a variety of constructions that given such polynomials can create other ones with the same property among these are algebraic analogues of the matroid operations of deletion contraction direct sum parallel connection series connection and 2 sum the complete monotonicity of p beta for some beta 0 can be viewed as a strong quantitative version of the half plane property hurwitz stability for p and is also related to the rayleigh property for matroids

generalized widder theorem via fractional moments abstract we provide a necessary and sufficient condition for the representability of a function as the classical multidimensional laplace transform when the support of the representing measure is contained in some generalized semi algebraic set this is done by employing a method of putinar and vasilescu m putinar f h vasilescu solving moment problems by dimensional extension ann of math 2 149 3 1999 1087 1107 for the corresponding multidimensional moment problem

hp submeshing via non conforming finite element methods abstract non conformity in the hp version can involve incompatibility in both the degrees and the meshes between adjoining subdomains in this paper we show how the mortar finite element method m0 and two new variants m1 m2 can be used to join together such incompatible hp sub discretizations our results show optimality of the resulting non conforming method for various h p and hp discretizations including the case of exponential hp convergence over geometric meshes we also present numerical results for the lagrange multiplier when the method is implemented via a mixed method three dimensional considerations suggest that our methods m1 m2 are easier to generalize to arbitrary meshes than m0

an explicit a posteriori error indicator for electromagnetic finite element boundary integral analysis within the finite element method fem there is always an error associated with the solution quantifying this error is a very important aspect of fe analysis since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost this paper presents an a posteriori error indicator for time harmonic electromagnetic analysis in three dimensions when using a galerkin fe approach with curl conforming elements and including a nonlocal boundary integral boundary condition as well as a dominant mode port boundary condition the error indicator is residual based of an explicit nature and constitutes a bound on an error energy measure the error indicator is evaluated by presenting results relating to common microwave engineering applications which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

a hierarchical iterative procedure for the analysis of composite laminates a multilevel recursively defined preconditioner for use with the preconditioned conjugate gradient pcg algorithm is developed and applied to solve thin and thick laminated plate problems the preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element fe method numerical studies have been conducted using quadrilateral elements for two dimensional problems up to p 8 and serendipity brick elements for three dimensional problems up to p 5 the efficiency of the iterative procedure is illustrated using bending response of laminated plates the effects of element span ratios convergence tolerance polynomial order material properties and plate span ratios on the number of iterations required for convergence has been investigated results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two and three dimensional problems in structural mechanics

the rogers ramanujan identities lie theoretic interpretation and proof abstract the two rogers ramanujan identities which equate certain infinite products with infinite sums are among the most intriguing of the classical formal power series identitites it has been found by lepowsky and milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the euclidean kac moody lie algebra a1 1 on the other hand the present authors have introduced an infinite dimensional heisenberg subalgebra unk of a1 1 which leads to a construction of a1 1 in terms of differential operators given by the homogeneous components of an exponential generating function in the present announcement we use unk to formulate a natural abstract rogers ramanujan identity for an arbitrary standard a1 1 module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules the abstract identity equates two expressions one a product and the other a sum for the principally specialized character of the space of highest weight vectors or vacuum states for unk in the module the construction of a1 1 leads to a concrete realization of as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions the summands in the rogers ramanujan identities turn out to count the dimensions of these spaces for general standard a1 1 modules we conjecture that the abstract identities agree with generalizations of the rogers ramanujan identities due to gordon andrews and bressoud

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

optimal h p finite element methods abstract a theory of optimal petrov galerkin h p version finite element approximations is presented the optimal scheme is denned relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form the optimal method leads to a symmetric positive definite stiffness matrix which is independent of the coefficients of the given problem exhibits extra superconvergence properties and has a relative error that can be calculated exactly at each point in the problem domain various generalizations are also discussed including the connection of these methods with certain preconditioning schemes

rogers ramanujan type identities and nil daha abstract using the daha fourier transform of q hermite polynomials multiplied by level one theta functions we obtain expansions of products of any number of such theta functions in terms of the q hermite polynomials an ample family of modular functions satisfying rogers ramanujan type identities for arbitrary reduced twisted affine root systems is obtained as an application a relation to rogers dilogarithm and nahm s conjecture is discussed the q hermite polynomials are closely related to the demazure level one characters in the twisted case sanderson ion which connects our formulas to tensor products of level one integrable kac moody modules their coset theory and the level rank duality

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

a global solution to the schrodinger equation from henstock to feynman one of the key elements of feynman s formulation of non relativistic quantum mechanics is a so called feynman path integral it plays an important role in the theory but it appears as a postulate based on intuition rather than a well defined object all previous attempts to supply feynman s theory with rigorous mathematics underpinning based on the physical requirements have not been satisfactory the difficulty comes from the need to define a measure on the infinite dimensional space of paths and to create an integral that would possess all of the properties requested by feynman in the present paper we consider a new approach to defining the feynman path integral based on the theory developed by muldowney a modern theory of random variable with applications in stochastic calcolus financial mathematics and feynman integration john wiley sons inc new jersey 2012 muldowney uses the henstock integration technique and deals with non absolute integrability of the fresnel integrals in orde

reflection positivity and conformal symmetry abstract a reflection positive hilbert space is a triple e e where e is a hilbert space a unitary involution and e a closed subspace on which the hermitian form v w v w is positive semidefinite for a triple g s where g is a lie group an involutive automorphism of g and s a subsemigroup invariant under the involution s s s 1 a unitary representation of g on e e is called reflection positive if g g and s e e this is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors this approach is most natural to obtain classification results in particular in the abelian case among the tools we develop is a generalization of the bochner schwartz theorem to positive definite distributions on open convex cones we further illustrate our techniques with a non abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group o 1 n 1 r of the sphere s n

application of hp adaptive be fe methods to elastic scattering abstract the investigation focuses on the development of hp adaptive methods for simulation of acoustical scattering problems the present formulation is based on steady state linear elasticity equations coupled with the burton miller integral equation representing the behavior of a surrounding acoustical medium the highlights of the proposed approach include a mathematically consistent variational formulation of the problem an hp adaptive fe be approximation based on standard galerkin methods an element by element solver a residual based a posteriori error estimate an hp adaptive strategy particular attention is given to asymptotic convergence and to a comparison of theory and numerical simulations

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

the h p adaptive finite element method for the numerical simulation of compressible flow an h p adaptive finite element method is presented for the numerical simulation of compressible fluid flow in addition to refining the mesh in regions where the estimated error is large the polynomial degree of the elements is increased it is shown that by using high order polynomials highly accurate approximations are obtained to a regularized form of the equations governing viscous compressible flow

solution of elastic scattering problems in linear acoustics using h p boundary element method abstract the work is a continuation of 1 a simplified 3d model problem reduced from the general class of elastic scattering problems is analyzed focusing attention on such practical details of the 3d implementation as a triangles based h p approximation and corresponding data structure integration of singular integrals and its effect on theoretically optimal rates of convergence a posteriori error estimation and sample adaptive strategies

free vibration analysis of membranes using the h p version of the finite element method abstract the h p version of the finite element method is applied to the vibration of membranes this is accomplished using a polynomially enriched triangular element new simple expressions of hierarchical c 0 shape functions for triangles are given in terms of the shifted legendre orthogonal polynomials the h p version of the finite element method marries both the concepts of the conventional h version and the p version the accuracy of the solution is sought by simultaneously refining the mesh and increasing the polynomial order in each element results of frequency calculations are found for triangular and l shaped membranes using a number of meshes and polynomial orders it is shown that the h p version of the finite element method is always convergent from above and produces a high accuracy with few degrees of freedom

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

convergence and optimality of mathbf hp hp afem we design and analyze an adaptive hp finite element method mathbf hp hp afem in dimensions n 1 2 n 1 2 the algorithm consists of iterating two routines mathbf hp hp nearbest finds a near best hp approximation of the current discrete solution and data to a desired accuracy and reduce improves the discrete solution to a finer but comparable accuracy the former hinges on a recent algorithm by binev for adaptive hp approximation and acts as a coarsening step we prove convergence and instance optimality

adaptive mesh generation for fluid mechanics problems in this study we give an in depth investigation of the adaptive procedures for compressible and incompressible flow problems many tests have been carried out with different error indicators for different problems these refinement procedures show that the interpolation error indicators are adequate to predict an accurate solution comparison between the gradient and curvature based error indicators show that the former yields better performance in most situations however we found that the latter indicators predict better motion in the recirculatory and unstable regions than the gradient based method many test cases have been presented for both compressible and incompressible flow problems in the later part of the paper we suggest a simple way of combining the curvature gradient indicators of different key variables and present an appropriate problem where it is applicable a wide coverage of relevant literature has also been presented in this paper copyright 2000 john wiley sons ltd