subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials this paper addresses the use of isogeometric analysis to solve solid mechanics problems involving nearly incompressible materials the present work is focused on extension of two field mixed variational formulations in both small and large strains to isogeometric analysis inf sup stable displacement pressure combinations for mixed formulations are developed based on the subdivision property of nurbs stability and convergence properties of the proposed displacement pressure combinations are illustrated by computing numerical inf sup constants and error norms the performance of the proposed formulations is assessed by studying several benchmark examples involving nearly incompressible and incompressible elastic and elasto plastic materials in both small and large strain regime

central limit theorems for infinite series of queues and applications to simple exclusion we prove that a tagged particle in asymmetric simple exclusion satisfies a central limit theorem when properly rescaled to obtain this result we derive several results of positive or negative correlation for occupation times of a server in a series of queues which imply various central limit theorems

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

uniform preconditioners for a parameter dependent saddle point problem with application to generalized stokes interface equations we consider an abstract parameter dependent saddle point problem and present a general framework for analyzing robust schur complement preconditioners the abstract analysis is applied to a generalized stokes problem which yields robustness of the cahouet chabard preconditioner motivated by models for two phase incompressible flows we consider a generalized stokes interface problem application of the general theory results in a new schur complement preconditioner for this class of problems the robustness of this preconditioner with respect to several parameters is treated results of numerical experiments are given that illustrate robustness properties of the preconditioner

diffusive limit of a tagged particle in asymmetric simple exclusion processes invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d 3 the method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions c 2000 john wiley sons inc

a unified mixed finite element approximations of the stokes darcy coupled problem abstract in this paper we develop and analyze a unified approximation of the velocity pressure pair for the stokes darcy coupled problem in a plane domain it is well known that stable finite element approximations for the stokes problem may not be appropriate for darcy problem and for the coupling of fluid flow modeled by the stokes equations with porous media flow modeled by the darcy equation and therefore different spaces are commonly used for the discretizations of the darcy and the stokes problems in this work we proposed a modification of the darcy problem which allows us to apply the classical mini element to the whole coupled stokes darcy problem the proposed method is probably one of the cheapest method for continuous approximation of the coupled system has optimal accuracy with respect to solution regularity and has simple and straightforward implementations numerical experiments are also presented which confirm the excellent stability and accuracy of our method

current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition for the symmetric simple exclusion process on an infinite line we calculate exactly the fluctuations of the integrated current qt during time t through the origin when in the initial condition the sites are occupied with density a on the negative axis and with density b on the positive axis all the cumulants of qt grow like sqrt t in the range where q t sim sqrt t the decay exp qt3 t of the distribution of qt is non gaussian our results are obtained using the bethe ansatz and several identities derived recently by tracy and widom for exclusion processes on the infinite line

a new nonconforming galerkin scheme for the stokes problem partially circumventing the discrete babǔska brezzi condition abstract in this paper we introduce a new stable nonconforming galerkin scheme for the interior stokes problem our approach uses a dirichllety mapping d from the pressure space q l 0 2 into the velocity space h h 0 1 2 in order to make a suitable change of variables in the original mixed formulation this transformation yields an equivalent variational formulation whose associated bilinear form is strongly coercive on the product space h q we then replace the mapping d by its finite element approximation on a subspace h h of h and define the corresponding discrete scheme on a finite element subspace h h q h of h q the leads to a nonconforming galerkin method whose bilinear form is uniformly strongly coercive on any family h h q h h h 0 provided that h o h consequently although h o h implies the satisfaction of the discrete bab ska brezzi condition between h h and q h it turns out that this stability condition is not needed between the approximating spaces h h and q h therefore all the combinations of velocity and pressure subspaces with continuous lagrange elements including certainly equal order interpolations become stable in addition the cea estimate is valid for all h h 0 and hence the usual rates of convergence for the error are also derived finally we discuss the matrix formulation of the algorithm and show that the resulting galerkin equations reduce to two uncoupled linear systems whose stiffness matrices are positive definite

on rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction this paper deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements we analyze the structure of the equations of motion and show that a differential algebraic system of index 1 with a singular perturbation term needs to be solved in the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem for the time integration we apply rosenbrock methods and study their convergence behavior for a test problem which highlights the importance of the well known scholz conditions for this problem class numerical tests demonstrate that such linear implicit methods are an attractive alternative to established time integration methods in structural dynamics we also combine the simulation of nonlinear materials with a model reduction step in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications

randomized polynuclear growth with a columnar defect we study a variant of poly nuclear growth where the level boundaries perform continuous time discrete space random walks and study how its asymptotic behavior is affected by the presence of a columnar defect on the line we prove that there is a non trivial phase transition in the strength of the perturbation above which the law of large numbers for the height function is modified

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

an anisotropic pressure stabilized finite element method for incompressible flow problems we consider a pressure stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios

on two types of obstructed random walks abstract two examples of random walks on a lattice which are subject to obstructing conditions are discussed i a random walk on a d dimensional lattice containing a periodic array of traps ii a random walk in a set of colliding random walks in d 1 expressions are derived for the probability distribution of the lattice point where the walk comes to an end and the average total number of distinct lattice points visited in example i and for the probability distribution of a walker s displacement in particular the mean square displacement for long times in example ii

tracer diffusion in lattice gases it has been proved that a tracer particle in a reversible lattice gas converges to brownian motion however only in a few particular cases has a strictly positive self diffusion coefficientd been established here we supply the missing piece and show thatd 0 in general the exceptions are one dimensional lattice gases with nearest neighbor jumps only for whichd 0 the proof establishes a variational formula ford which could be used to obtain realistic bounds

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

self diffusion in simple models systems with long range jumps we review some exact results for the morion of a tagged particle in simple models then we study the density dependence of the sill diffusion coefficient d n in lattice systems with simple symmetric exclusion in which the particles can jump with equal rates to a set of n neighboring sites we obtain positive upper and lower bounds on f n n 1 d n d n 0 1 for is an element of 0 1 computer simulations for the square triangular and one dimensional lattices suggest that fn becomes effectively independent of n for n greater than or equal to 20

inf sup stable non conforming finite elements of arbitrary order on triangles we introduce a family of scalar non conforming finite elements of arbitrary order k 1 with respect to the h1 norm on triangles their vector valued version generates together with a discontinuous pressure approximation of order k 1 an inf sup stable finite element pair of order k for the stokes problem in the energy norm for k 1 the well known crouzeix raviart element is recovered

incompressible navier stokes solutions by a triangular spectral p element projection method the application of the fractional step projection method recently proposed by guermond and quartapelle to the numerical approximation of unsteady navier stokes solutions by means of a spectral p element method is considered in particular we illustrate the second order pressure correction technique and evaluate its accuracy properties in some test cases stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed the high spectral accuracy in space and the second order accuracy in time are verified by two simple test cases with analytical solution a more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries

a mixed finite element method for nearly incompressible multiple network poroelasticity in this paper we present and analyze a new mixed finite element formulation of a general family of quasi static multiple network poroelasticity mpet equations the mpet equations describe flow a

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

a limit theorem for financial markets with inert investors we study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time it turns out that semi markov processes are tailor made for modelling inert investors with a suitable scaling we show that when the price is driven by the market imbalance the log price process is approximated by a process with long range dependence and non gaussian returns distributions driven by a fractional brownian motion consequently investor inertia may lead to arbitrage opportunities for sophisticated market participants the mathematical contributions are a functional central limit theorem for stationary semi markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional brownian motion

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

exclusion process for particles of arbitrary extension hydrodynamic limit and algebraic properties the behaviour of extended particles with exclusion interaction on a one dimensional lattice is investigated the basic model is called l asep as a generalization of the asymmetric exclusion process asep to particles of arbitrary length l stationary and dynamical properties of the l asep with periodic boundary conditions are derived in the hydrodynamic limit from microscopic properties of the underlying stochastic many body system in particular the hydrodynamic equation for the local density evolution and the time dependent diffusion constant of a tracer particle are calculated as a fundamental algebraic property of the symmetric exclusion process the su 2 symmetry is generalized to the case of extended particles

shock fluctuations in the asymmetric simple exclusion process we consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively q p ferrari et al 1991 have shown that if the initial measure isv a product measure with densities and to the left and right of the origin respectively then there exists a microscopic shock for the system a shock is a random positionx t such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively uniformly int we compute the diffusion coefficient of the shockd lim t t 1 e x t 2 ex t 2 and findd p q 1 1 1 as conjectured by spohn 1991 we show that in the scale sqrt t the position ofx t is determined by the initial distribution of particles in a region of length proportional tot we prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and this is the so called dynamical phase transition under shock initial conditions we show how the density fluctuation fields depend on the initial configuration

cumulant expansions for thermal density matrices and ground state logarithmic perturbation series we express the logarithm of the thermal density matrix h x x x e h x where h h0 v as a cumulant expansion in powers of the perturbation v by associating with h0 x x a stochastic process if the ground state of h0 is isolated this stochastic process is of finite memory it then follows from the properties of cumulants that the above expansion of ln is nonsecular as all its terms and is thus usable down to zero temperature unlike the direct expansion of in powers of v whose nth term n to determine explicitly the low temperature behavior we apply an analysis familiar in the theory of relaxation and obtain the form ln h x x h x x a x a x b where a b h are cumulant expansions in powers of v a x b are independent of and h x x 0 as comparing with x 0 e e0 0 x as where 0 and e0 are the ground state and energy of h we deduce e0 b ln x 0 a x i e the rayleigh schrodinger perturbation series for e0 and ln x 0 with 0 0

adaptive inexact iterative algorithms based on polynomial degree robust a posteriori estimates for the stokes problem in this paper we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear stokes problem we base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components namely the discretization error component the inner algebraic solver error component and possibly the outer algebraic solver error component for algorithms of the uzawa type we prove that our estimate gives a guaranteed upper bound on the total error as well as a polynomial degree robust local efficiency and this on each step of the employed iterative algorithm our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error the developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver implementation into the freefem programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies as example we choose here the unpreconditioned and preconditioned uzawa algorithm and the preconditioned minimum residual minres algorithm in combination with the taylor hood discretization

quantum virial coefficients as cumulants of imaginary time ordered mayer diagrams the virial coefficients for a quantum gas including quantum statistics are expressed as sums of cumulants of connected generalized mayer diagrams the cumulants being built on the irreducible blocks of the diagrams the mayer diagrams are defined for the quantum case in terms of imaginary time ordered exponentials the quantum statistics being incorporated in the guise of multiparticle interactions in order to extend mayer diagrams to multiparticle interactions we utilize terminology and methods from the theory of hypergraphs the virial coefficients naturally separate into a quantum boltzmann gas contribution an ideal quantum gas contribution and a final term expressing correlations between dynamics and statistics in the classical limit connected mayer diagrams factorize into their irreducible blocks the cumulants over irreducible blocks then vanish by a basic property of cumulants except for diagrams which are themselves irreducible whence the classical result of mayer extended to multip

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

regularity of quasi stationary measures for simple exclusion in dimension d 5 we consider the symmetric simple exclusion process on zz d for d geq 5 and study the regularity of the quasi stationary measures of the dynamics conditioned on not occupying the origin for each rho in 0 1 we establish uniqueness of the density of quasi stationary measures in l 2 d nur where nur is the stationary measure of density rho this in turn permits us to obtain sharp estimates for p nur tau t where tau is the first time the origin is occupied

t e harris contributions to interacting particle systems and percolation this is a bird s eye view of t e harris work on interacting particle systems and percolation and of its impact on later work in probability theory and mathematical physics nal illness and death was a great loss to me personally and professionally as well as to probability theory in southern california and beyond for three decades he and i were the primary probabilists at usc and ucla respectively our approach to mathematics and our mathematical interests were very similar my wife chris and i enjoyed many wonderful social occasions at the harris home in beverly hills with ted and connie they were great hosts the conversation was always stimulating they were interested in so many things ted initiated the southern california probability symposium in about 1970 it is probably the oldest meeting of its type in the us and continues to provide an exciting place for interactions among southern california probabilists to this day in two of the years 1989 and 2007 the meeting was dedicated to ted the rst was on the occasion of his 70th birthday and the second was in his memory even after his retirement ted attended the usc probability seminar regularly always sitting in the front row and asking perceptive questions few mathematicians have had a greater ratio of number of ideas to number of papers ted wrote many fewer papers 31 than most prominent mathematicians but each is a jewel in each of the several areas in which he worked he was among the

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

cumulants and the moment algebra tools for analyzing weak measurements recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements here we consider the mathematical structure that underlies this and find that it can be formulated in terms of what we call the moment algebra apart from resulting in simpler proofs the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios including one where the weakly interacting pointers reach thermal equilibrium with the probed system

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

clustering in the one dimensional three color cyclic cellular automaton cyclic cellular automaton the author has previously shown that this process fluctuates meaning that each lattice site changes color infinitely often so that there is no final state for the system the focus of the current work is on the clustering properties of this system this paper demonstrates that the one dimensional three color cyclic cellular automaton clusters and the mean cluster size as a function of time t is asymptotic to ct1 2 where c is an explicitly calculable constant the method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one dimensional system of particles which undergo deterministic motion and which annihilate upon collision no clustering results are known about the four color process but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from t1 2

the tasep speed process in the multi type totally asymmetric simple exclusion process tasep each site ofz is occupied by a labeled particle and two neighboring particles are int erchanged at rate one if their labels are in increasing order consider the process with the initial con figuration where each particle is labeled by its position it is known that in this case a s each particle has an asymptotic speed which is distributed uniformly on 1 1 we study the joint distribution of these speeds the tasep speed process we prove that the tasep speed process is a stationary measure for the multi type tasep dynamics consequently we can describe every ergodic stationary measure as a projection of the speed process measure this generalizes previous descriptions restricted to finitely many classes by combining this result with known stationary measures for taseps with finitely many types we compute several marginals of the speed process including the joint density of two and three consecutive speeds one striking property of the distribut ion is that two speeds are equal with positive probability and for any given particle there are infinitely m any others with the same speed we also study the partially asymmetric simple exclusion process asep we prove that the states of the asep with the above initial configuration seen as permutations ofz are symmetric in distribution this allows us to extend some of our results i ncluding the stationarity and description of all ergodic stationary measures also to the asep

on the locking of the finite element method in thermoelasticity in this article we discuss the solution u to the stationary problem of thermoelasticity and the approximation un of u obtained with a standard galerkin method in the incompressible limit when the lame coefficient lambda becomes infinite un is subject to a divergence constraint div un pnt where pn is a given approximation operator acting on the temperature field t this mechanism provokes a locking phenomenon similar to the one observed in elasticity and studied by babuska and suri in numer math 62 1992 pp 439 464 the purpose of this work is to give a sufficient and necessary condition on the discrete spaces vn used in the galerkin formulation to avoid locking and to get an estimation of the error u un 1 which is optimal and independent of lambda this criterion will be used to construct two practical examples of methods free from locking for the usual thermoelasticty problem where the temperature field is the solution of a poisson equation the first example is based on the h version of the finite element method the second one on the p version

multi species simple exclusion processes abstract a motility mechanism based on a simple exclusion process where the movement of discrete agents on a lattice is either unbiased symmetric or biased asymmetric is considered estimates of diffusivities from tracking data do not describe the population level response of the system this mismatch between the individual level and population level behaviour can be resolved by averaging the individual level mechanism in terms of an expected site occupancy new insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations this formalism leads to a system of nonlinear advection diffusion equations which can be interpreted in terms of the agent fluxes these interactions have consequences for both agent based modelling and continuum modelling in cell biology such as tracking subpopulations of cells within a total cell population

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

equilibrium fluctuation of the atlas model we study the fluctuation of the atlas model where a unit drift is assigned to the lowest ranked particle among a semi infinite z z indexed system of otherwise independent brownian particles initiated according to a poisson point process on r r in this context we show that the joint law of ranked particles after being centered and scaled by t 14t 14 converges as t t to the gaussian field corresponding to the solution of the additive stochastic heat equation ashe on r r with the neumann boundary condition at zero this allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional brownian motion fbm in particular we prove a conjecture of pal and pitman ann appl probab 18 2008 2179 2207 about the asymptotic gaussian fluctuation of the ranked particles

power law polya s urn and fractional brownian motion we introduce a natural family of random walks s n on mathbb z that scale to fractional brownian motion the increments x n s n s n 1 in pm 1 have the property that given x k k n the conditional law of x n is that of x n k n where k n is sampled independently from a fixed law mu on the positive integers when mu has a roughly power law decay precisely when mu lies in the domain of attraction of an alpha stable subordinator for 0 alpha 1 2 the walks scale to fractional brownian motion with hurst parameter alpha 1 2 the walks are easy to simulate and their increments satisfy an fkg inequality in a sense we describe they are the natural fractional analogues of simple random walk on mathbb z

on some partitions of hypergraphs and cumulants having applications in statistical mechanics a hypergraph h e1 e2 en is a set of sets the sets ei are called the edges and the elements of v h uni 1ei the vertices of h if each edge contains only two vertices we have an ordinary graph hypergraphs and cumulants over hypergraphs occur in various statistical mechanical problems a cumulant over h involves a sum over partitions of the set h whence the motivation for studying partitions of hypergraphs and related notions writing ei k v ej if there exist at least k n vertex disjoint paths between the edges ei and ej we show that k v is an equivalence relation on h and denote by pkh the partition of h into its k v equivalence classes h is said to be k v connected if every pair of edges is in the relation k v we show that there exists a coarsest partition of h into k v connected blocks and denote it by p kh given a partition p h h1 h2 hp of h we denote for any real number p h pj 1 v hj p h pj 1 v hj mkp1hj

current fluctuations of a system of one dimensional random walks in random environment we study the current of particles that move independently in a common static random environment on the one dimensional integer lattice a two level fluctuation picture appears on the central limit scale the quenched mean of the current process converges to a brownian motion on a smaller scale the current process centered at its quenched mean converges to a mixture of gaussian processes these gaussian processes are similar to those arising from classical random walks but the environment makes itself felt through an additional brownian random shift in the spatial argument of the limiting current process

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

a posteriori error estimations for mixed finite element approximations to the navier stokes equations based on newton type linearization abstract in this paper we derive a posteriori error estimates for inf sup stable mixed finite element approximations to the evolutionary navier stokes equations we reduce the problem of getting a posteriori error estimations of a non linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem the main idea is based on the fact that the solutions of some newton type linearized navier stokes equations around the plain galerkin approximations approach the solution of the original navier stokes equations with a bigger rate of convergence than the plain galerkin method as a consequence the difference between the galerkin approximations and the solution of the linearized problem is an estimator of the galerkin error moreover since the galerkin approximation of the evolutionary navier stokes equations is also the galerkin approximation of the linearized equations any a posteriori estimator of the error in the newton type linearized navier stokes equations is also an a posteriori error estimator of the full navier stokes equations

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

generalized postprocessed approximations to the navier stokes equations based on two grids abstract in this paper we generalize previous postprocessed approximations to the navier stokes equations the idea of postprocessing in the context of mixed finite element approximations is the following once a standard galerkin mixed finite element approximation is computed using a mesh we will call coarse of size h at a fixed time t we postprocess the approximation using a finer mesh of size h h by solving a steady problem with data based on the galerkin approximation in the literature one can find different ways to postprocess the galerkin approximation depending on the problem that is solved over the fine mesh at the final time t basically one can solve a stokes problem an oseen type problem or a newton type problem in the present paper we present a method based on several parameters that generalize the above postprocessing techniques depending on the values chosen for the different parameters one can recover one of the old known postprocessing procedures but also produce a new different method we get error bounds for the generalized method valid for any of the values of the different parameters in all the cases the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain galerkin method in terms of the coarse mesh h and optimal in terms of the fine mesh h the computational added cost of postprocessing is however negligible for the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the navier stokes equations some numerical experiments show the performance of the method for different values of the parameters

diffusion of finite sized hard core interacting particles in a one dimensional box tagged particle dynamics we solve a nonequilibrium statistical mechanics problem exactly namely the single file dynamics of n hard core interacting particles the particles cannot pass each other of size delta diffusing in a one dimensional system of finite length l with reflecting boundaries at the ends we obtain an exact expression for the conditional probability density function rho t yt t vertical bar yt 0 that a tagged particle t t 1 n is at position yt at time t given that it at time t 0 was at position yt 0 using a bethe ansatz we obtain the n particle probability density function and by integrating out the coordinates and averaging over initial positions of all particles but particle t we arrive at an exact expression for rho t yt t vertical bar yt 0 in terms of jacobi polynomials or hypergeometric functions going beyond previous studies we consider the asymptotic limit of large n maintaining l finite using a nonstandard asymptotic technique we derive an exact expression for rho t yt t vertical bar yt 0 for a tagged particle located roughly in the middle of the system from which we find that there are three time regimes of interest for finite sized systems a for times much smaller than the collision time t tau coll 1 rho d 2 where rho n l is the particle concentration and d is the diffusion constant for each particle the tagged particle undergoes a normal diffusion b for times much larger than the collision time t tau coll but times smaller than the equilibrium time t tau eq l 2 d we find a single file regime where rho t yt t vertical bar yt 0 is a gaussian with a mean square displacement scaling as t 1 2 and c for times longer than the equilibrium time t tau eq rho t yt t vertical bar yt 0 approaches a polynomial type equilibrium probability density function notably only regimes a and b are found in the previously considered infinite systems

some properties of k step exclusion processes we introduce k step exclusion processes as generalizations of the simple exclusion process we state their main equilibrium properties when the underlying stochastic matrix corresponds to a random walk or is positive recurrent and reversible finally we prove laws of large numbers for tagged and second class particles

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection abstract in this paper we present a stabilized finite element formulation for the transient incompressible navier stokes equations the main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the laplacian of the pressure and the divergence of this new vector field this leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures in the case of the transient equations we consider the possibility of treating the pressure gradient projection either implicitly or explicity in the first case the number of unknowns of the problem is substantially increased with respect to the standard galerkin formulation nevertheless iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns velocity and pressure when this vector field is treated explicitly the increment of computational cost of the stabilized formulation with respect to the galerkin method is very low we provide a stability estimate for the case of the simple backward euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection

meteor process on mathbb z d the meteor process is a model for mass redistribution on a graph the case of finite graphs was analyzed in billey et al on meteors earthworms and wimps ann appl probab 2014 this paper is devoted to the meteor process on mathbb z d the process is constructed and a stationary distribution is found convergence to this stationary distribution is proved for a large family of initial distributions the first two moments of the mass distribution at a vertex are computed for the stationary distribution for the one dimensional lattice mathbb z the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity an alternative representation of the process on mathbb z as a collection of non crossing paths is presented the distributions of a tracer particle in this system of non crossing paths are shown to be tight as time goes to infinity

a simple construction of a fortin operator for the two dimensional taylor hood element abstract a fortin operator is constructed to verify the discrete inf sup condition for the lowest order taylor hood element and its variant in two dimensions the approach is closely related to the recent work by mardal et al 2013 that is based on the isomorphism of the tangential edge bubble function space to a subspace of the lowest order edge element space a more precise characterization of this subspace and a numerical quadrature are introduced to simplify the analysis and remove the mesh restriction the constructed fortin operator is stable in both h 1 and l 2 norm for general shape regular triangulations

stable q k q k 1 mixed finite elements with discontinuous pressure mixed finite elements play a central role in many important cfd applications involving stokes solvers and alike a natural mixed finite element for the stokes equations is the q k q k 1 element on rectangular grids by which the velocity is approximated by continuous polynomials of separated degree k and the pressure is approximated by discontinuous polynomials of separated degree k 1 such an element is however not stable we propose in this paper three modified q k q k 1 elements with certain element wise divergence free property of velocity where the pressure space is slightly restricted to subspaces in c 1 q k 1 yet the optimal order of approximation is still retained the stability and approximation analysis for the new elements are presented comprehensive numerical experiments are also conducted to confirm the theoretical analysis and to reveal the super convergence for some of these new elements