on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

the role of geometric stiffness in momentum and energy conserving time integration a momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic green strain momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level it is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix over the current time step usually directly available in global form in existing finite element programmes the theory is derived by the use of a state space formulation where this extra term is located in the same position as the viscous damping matrix indicating that the effect of the extra incremental geometric stiffness term in the non linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment in the actual numerical algorithm the new value of the velocity vector is eliminated leaving a non linear equation for the displacement increment alone followed by an explicit vector update of the velocity increment copyright 2006 john wiley sons ltd

integration of elastic multibody systems by invariant conserving dissipating algorithms ii numerical schemes and applications abstract this work presents a novel methodology for the dynamic analysis of general non linear multibody systems composed of rigid and deformable bodies the latter under the small strain assumption in part i we developed the 6 d compact representation and parameterization of motion for constrained bodies part ii is devoted to the design of a class of modified runge kutta rk methods dedicated to non linear dynamics these are capable of integrating on the configuration manifold and of preserving linear and angular momenta within this class of methods two second order algorithms are designed under the requirement of attaining non linear unconditional stability the energy preserving ep and energy decaying ed methods these schemes are associated with an algorithmic law of conservation and dissipation respectively of the total mechanical energy of the system together with the vanishing of the algorithmic work done by ideal time independent constraints their performances are assessed with the aid of some representative numerical applications which confirm the non conventional properties predicted in the analysis

sublinear variance in first passage percolation for general distributions we prove that the variance of the passage time from the origin to a point x in first passage percolation on mathbb z d is sublinear in the distance to x when d ge 2 obeying the bound c vert x vert log vert x vert under minimal assumptions on the edge weight distribution the proof applies equally to absolutely continuous discrete and singular continuous distributions and mixtures thereof and requires only 2 log moments the main result extends work of benjamini kalai schramm ann prob 31 2003 and benaim rossignol ann inst henri poincare prob stat 3 2008

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

geometrically exact intrinsic theory for dynamics of curved and twisted anisotropic beams a formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams when the applied loads at the ends of and distributed along the beam are independent of the deformation neither displacement nor rotation variables appear an intrinsic formulation like well known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two advantages of such a formulation are demonstrated with a simple example when the initial curvature and twist are constant along the beam two space time conservation laws are shown to exist one being a work energy relation and the other a generalized impulse momentum relation these laws can be used for example as benchmarks to check the accuracy of any proposed solution including time marching and finite element schemes the structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization a particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end loaded beams that by virtue of the kirchhoff analogy leads to a time marching scheme for the dynamics of a pivoted rigid body in a gravity field this time marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy whereas the analogous spatial discretization scheme for the nonlinear static behavior of end loaded beams satisfies analogous integrals of deformation along the beam span remarkably a straightforward generalization of these discretization schemes is shown to satisfy both space time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space time element

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

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

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

energy momentum consistent finite element discretization of dynamic finite viscoelasticity this paper is concerned with energy momentum consistent time discretizations of dynamic finite viscoelasticity energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics the discretization is energy momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations the performed approximation is based on a three field formulation in which the deformation field the velocity field and a strain like viscous internal variable field are treated as independent quantities the new non linear viscous evolution equation satisfies a non negative viscous dissipation not only in the continuous case but also in the fully discretized system the initial boundary value problem is discretized by using finite elements in space and time thereby the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments incremental objectivity although the present approach makes possible to design schemes of arbitrary order the focus is on finite elements relying on linear lagrange polynomials for the sake of clearness the discrete energy momentum consistency is based on the collocation property and an enhanced second piola kirchhoff stress tensor the obtained coupled non linear algebraic equations are consistently linearized the corresponding iterative solution procedure is associated with newly proposed convergence criteria which take the discrete energy consistency into account the iterative solution procedure is therefore not complicated by different scalings in the independent variables since the motion of the element is taken into account for solving the viscous evolution equation representative numerical simulations with various boundary conditions show the superior stability of the new time integration algorithm in comparison with the ordinary midpoint rule both the quasi rigid deformations during a free flight and large deformations arising in a dynamic tensile test are considered copyright 2009 john wiley sons ltd

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

recursive supervised estimation of a markov switching garch process in the short time fourier transform domain in this paper we introduce a markov switching generalized autoregressive conditional heteroscedasticity garch model for nonstationary processes with time varying volatility structure in the short time fourier transform stft domain the expansion coefficients in the stft domain are modeled as a multivariate complex garch process with markov switching regimes the garch formulation parameterizes the correlation between sequential conditional variances while the markov chain allows the process to switch between regimes of different garch formulations we obtain a necessary and sufficient condition for the asymptotic wide sense stationarity of the model and develop a recursive algorithm for signal restoration in a noisy environment the conditional variance is estimated by iterating propagation and update steps with regime conditional probabilities while the model parameters are evaluated a priori from a training data set experimental results demonstrate the performance of the proposed algorithm

results on finite wireless networks on a line analysis of finite wireless networks is a fundamental problem in the area of wireless networking today due to the vast amount of literature on large scale wireless networks we have a fair understanding of the asymptotic behavior of such networks however in real world we have to face finite networks for which the asymptotic results cease to be valid we refer to networks as being finite when the number of nodes is less than a few hundred here we study a model of wireless networks represented by random geometric graphs in order to address a wide class of the network s properties we study the threshold phenomena being extensively studied in the asymptotic case the threshold phenomena occurs when a graph theoretic property such as connectivity of the network experiences rapid changes over a specific interval of the underlying parameter here we find an upper bound for the threshold width of finite line networks represented by random geometric graphs these bounds hold for all monotone properties of such networks we then turn our attention to an important non monotone characteristic of line networks which is the medium access mac layer capacity i e the maximum number of possible concurrent transmissions towards this goal we provide a linear time algorithm which finds a maximal set of concurrent non interfering transmissions and further derive lower and upper bounds for the cardinality of the set using simulations we show that these bounds serve as reasonable estimates for the actual value of the mac layer capacity

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

integration of elastic multibody systems by invariant conserving dissipating algorithms i formulation abstract this work presents a novel methodology for the dynamic analysis of general non linear flexible multibody systems in part i we develop the 6 d compact representation of motion for those body models which motion may be described by a displacement field plus an independent rotation field this approach explores the fundamental properties of rigid body motion and in particular the coupled nature of linear and angular quantities in both kinematics and dynamics inspiring a novel parameterization technique based on the exponential map using the proposed approach we derive the governing equations for the case of multibody systems composed by rigid bodies and geometrically non linear beams connected by holonomic constraints these equations provide the starting point for the derivation of a class of numerical algorithms characterized by non conventional conservation properties in part ii of this work we develop the algorithms and illustrate their properties with the aid of some numerical applications

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

the strain based beam finite elements in multibody dynamics we present a strain based finite element formulation for the dynamic analysis of flexible elastic planar multibody systems composed of planar beams we consider finite displacements rotations and strains the discrete dynamic equations of motion are obtained by the collocation method the strains are the basic interpolated variables which makes the formulation different from other formulations the further speciality of the formulation is the strong satisfaction of the cross sectional constitutive conditions at collocation points in order to avoid the nested integrations a special algorithm for the numerical integration over the length of the finite element is proposed the midpoint scheme is used for the time integration the performance of the formulation is illustrated via numerical examples including a stiff multibody system

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

approximate zero one laws and sharpness of the percolation transition in a class of models including two dimensional ising percolation one of the most well known classical results for site percolation on the square lattice is the equation p c p c 1 in words this equation means that for all values p c of the parameter p the following holds either a s there is an infinite open cluster or a s there is an infinite closed star cluster this result is closely related to the percolation transition being sharp below p c the size of the open cluster of a given vertex is not only a s finite but has a distribution with an exponential tail the analog of this result has been proven by higuchi in 1993 for two dimensional ising percolation at fixed inverse temperature c with external field h the parameter of the model using sharp threshold results approximate zero one laws and a modification of an rsw like result by bollobas and riordan we show that these results hold for a large class of percolation models where the vertex values can be nicely represented in a sense which will be defined precisely by i i d random variables we point out that the ordinary percolation model obviously belongs to this class and we also show that the ising model mentioned above belongs to it