conditional intensity and gibbsianness of determinantal point processes the papangelou intensities of determinantal or fermion point processes are investigated these exhibit a monotonicity property expressing the repulsive nature of the interaction and satisfy a bound implying stochastic domination by a poisson point process we also show that determinantal point processes satisfy the so called condition which is a general form of gibbsianness under a continuity assumption the gibbsian conditional probabilities can be identified explicitly

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

determinant permanent and macmahon s master theorem abstract as a unified extension of determinant and permanent a new matrix function with two extra parameters is introduced basic formulas are established which contain several known relations between determinants and permanents as special cases some expansions in terms of the matrix functions are investigated which yield macmahon s master theorem its dual version and their common generalization

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

white noise analysis for levy processes we construct a white noise theory for levy processes the starting point of this theory is a chaos expansion for square integrable random variables we use this approach to malliavin calculus to prove the following white noise generalization of the clark haussmann ocone formula for levy processes f e f m 1 0te dt m f ft y t m dt here e f is the generalized expectation the operators dt m f m 1 are generalized malliavin derivatives is the wick product and for all m 1y t m is the white noise of power jump processes yt m in particular y t 1 is the white noise of the levy process the formula holds for all f g l2 where g is a space of stochastic distributions and is a white noise probability measure finally we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by levy processes

universality properties of gelfand tsetlin patterns a standard gelfand tsetlin pattern of depth n is a configuration of particles in 1 ldots n times mathbb r for each r in 1 ldots n r times mathbb r is referred to as the rth level of the pattern a standard gelfand tsetlin pattern has exactly r particles on each level r and particles on adjacent levels satisfy an interlacing constraint probability distributions on the set of gelfand tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random hermitian matrices of size n we consider such probability spaces when the distribution of the matrix is unitarily invariant prove a determinantal structure for a broad subclass and calculate the correlation kernel in particular we consider the case where the eigenvalues of the random matrix are fixed this corresponds to choosing uniformly from the set of gelfand tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix fixing q n in 1 ldots n and letting n under the assumption that frac q n n to alpha in 0 1 and the empirical distribution of the particles on the nth level converges weakly the asymptotic behaviour of particles on level q n is relevant to free probability theory saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the sine kernel

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

periodic tasep with general initial conditions we consider the one dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain and obtain explicit formulas for the multi point distributions in the space time plane the formulas are given in terms of an integral involving a fredholm determinant we then evaluate the large time large period limit in the relaxation time scale which is the scale such that the system size starts to affect the height fluctuations the limit is obtained assuming certain conditions on the initial condition which we show that the step flat and step flat initial conditions satisfy hence we obtain the limit theorem for these three initial conditions in the periodic model extending the previous work on the step initial condition we also consider uniform random and uniform step random initial conditions

the malliavin calculus for pure jump processes and applications to local time on obtient trois resultats une formule d integration par parties des conditions pour qu un processus de markov de saut pur ait une densite en lp et des conditions pour qu une martingale purement discontinue ait un temps local

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

matrix free subcell residual distribution for bernstein finite element discretizations of linear advection equations abstract in this work we introduce a new residual distribution rd framework for the design of bound preserving high resolution finite element schemes the continuous and discontinuous galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing led approximations to that end we perform mass lumping and redistribute the element residuals in a manner which guarantees the led property the hierarchical correction procedure for high order bernstein finite element discretizations involves localization to subcells and definition of bound preserving weights for subcell contributions using strong stability preserving ssp runge kutta methods for time integration we prove the validity of discrete maximum principles under cfl like time step restrictions the low order version of our method has roughly the same accuracy as the one derived from a piecewise multi linear approximation on a submesh with the same nodal points in high order extensions we use an element based flux corrected transport fct algorithm which can be interpreted as a nonlinear rd scheme the proposed led corrections are tailor made for matrix free implementations which avoid the rapidly growing cost of matrix assembly for high order bernstein elements the results for 1d 2d and 3d test problems compare favorably to those obtained with the best matrix based approaches

energy image density property and the lent particle method for poisson measures abstract we introduce a new approach to absolute continuity of laws of poisson functionals it is based on the energy image density property for dirichlet forms the associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle and taking it back after some calculation

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

calculation of separation bubbles using a partially parabolized navier stokes procedure a partially parabolized navier stokes solution procedure for the transitional separation bubble has been developed by coupling with the far field potential flow solution the calculation domain can be kept relatively small to minimise the computation time using the eddy viscosity turbulence model of baldwin and lomax it is shown that the procedure is capable of capturing the details of the separation bubble the results are in excellent agreement with experimental data and exhibit a marked improvement over those of existing interactive boundary layer procedures for the cases examined

low complexity finite element algorithms for the de rham complex on simplices we combine recently developed finite element algorithms based on bernstein polynomials m ainsworth g andriamaro and o davydov siam j sci comput 33 2011 pp 3087 3109 r c kirby numer math 117 2011 pp 631 652 with the explicit basis construction of the finite element exterior calculus d n arnold r s falk and r winther comput methods appl mech engrg 198 2009 pp 1660 1672 to give a family of algorithms for the de rham complex on simplices that achieves stiffness matrix construction and matrix free action in optimal complexity these algorithms are based on realizing the exterior calculus bases as short combinations of bernstein polynomials numerical results confirm optimal scaling of the algorithms and favorable comparison with fenics at high polynomial order as well additional empirical studies show that very high accuracy is achieved in the mixed discretization of the poisson equation and the maxwell eigenvalue problem as the polynomial degree increases

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

scaling limit for the space time covariance of the stationary totally asymmetric simple exclusion process the totally asymmetric simple exclusion process tasep on the one dimensional lattice with the bernoulli measure as initial conditions 0 1 is stationary in space and time let nt j be the number of particles which have crossed the bond from j to j 1 during the time span 0 t for open image in new window we prove that the fluctuations of nt j for large t are of order t1 3 and we determine the limiting distribution function open image in new window which is a generalization of the gue tracy widom distribution the family open image in new window of distribution functions have been obtained before by baik and rains in the context of the png model with boundary sources which requires the asymptotics of a riemann hilbert problem in our work we arrive at open image in new window through the asymptotics of a fredholm determinant open image in new window is simply related to the scaling function for the space time covariance of the stationary tasep equivalently to the asymptotic transition probability of a single second class particle

on the distribution of surface extrema in several one and two dimensional random landscapes we study here a standard next nearest neighbor nnn model of ballistic growth on one and two dimensional substrates focusing our analysis on the probability distribution function p m l of the number m of maximal points i e local peaks of growing surfaces our analysis is based on two central results i the proof presented here of the fact that uniform one dimensional ballistic growth process in the steady state can be mapped onto rise and descent sequences in the ensemble of random permutation matrices and ii the fact established in ref g oshanin and r voituriez j phys a math gen 37 6221 2004 that different characteristics of rise and descent patterns in random permutations can be interpreted in terms of a certain continuous space hammersley type process for one dimensional system we compute p m l exactly and also present explicit results for the correlation function characterizing the enveloping surface for surfaces grown on 2d substrates we pursue similar approach considering the ensemble of permutation matrices with long ranged correlations determining exactly the first three cumulants of the corresponding distribution function we define it in the scaling limit using an expansion in the edgeworth series and show that it converges to a gaussian function as l

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

fast simplicial finite element algorithms using bernstein polynomials fast algorithms for applying finite element mass and stiffness operators to the b form of polynomials over d dimensional simplices are derived these rely on special properties of the bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor product techniques in rectangular domains first special structure leading to fast application of mass matrices is developed then by factoring stiffness matrices into products of sparse derivative matrices with mass matrices fast algorithms are also obtained for stiffness matrices

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

maximum distributions of bridges of noncolliding brownian paths one dimensional brownian motion starting from the origin at time t 0 conditioned to return to the origin at time t 1 and to stay positive during time interval 0 t 1 is called the bessel bridge with duration 1 we consider an n particle system of such bessel bridges conditioned never to collide with each other in 0 t 1 which is the continuum limit of the vicious walk model in watermelon configuration with a wall distributions of maximum values of paths attained in the time interval t epsilon 0 1 are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane for the outermost path the distribution function of maximum value is exactly determined for general n we show that the present n path system of noncolliding bessel bridges is realized as the positive eigenvalue process of the 2nx2n matrix valued brownian bridge in the symmetry class c using this fact computer simulations are performed and numerical results on the n dependence of the maximum value distributions of the inner paths are reported the present work demonstrates that the extreme value problems of noncolliding paths are related to random matrix theory the representation theory of symmetry and number theory

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

transversal fluctuations of the asep stochastic six vertex model and hall littlewood gibbsian line ensembles we consider the asep and the stochastic six vertex model started with step initial data after a long time t it is known that the one point height function fluctuations for these systems are of order t1 3 we prove the kpz prediction of t2 3 scaling in space namely we prove tightness and brownian absolute continuity of all subsequential limits as t goes to infinity of the height function with spatial coordinate scaled by t2 3 and fluctuations scaled by t1 3 the starting point for proving these results is a connection discovered recently by borodin bufetov wheeler between the stochastic six vertex height function and the hall littlewood process a certain measure on plane partitions interpreting this process as a line ensemble with a gibbsian resampling invariance we show that the one point tightness of the top curve can be propagated to the tightness of the entire curve

gibbsianness of fermion random point fields we consider fermion or determinantal random point fields on euclidean space d given a bounded translation invariant and positive definite integral operator j on l2 d we introduce a determinantal interaction for a system of particles moving on d as follows the n points located at x1 x n d have the potential energy given by where j x y is the integral kernel function of the operator j we show that the gibbsian specification for this interaction is well defined when j is of finite range in addition and for d 2 if the intensity is small enough we show that the fermion random point field corresponding to the operator j i j 1 is a gibbs measure admitted to the specification

tilings of non convex polygons skew young tableaux and determinantal processes this paper studies random lozenge tilings of general non convex polygonal regions we show that the pairwise interaction of the non convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations the precise geometrical figure here consists of a hexagon with cuts along opposite edges for this model we take limits when the size of the hexagon and the cuts tend to infinity while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit we show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral where the number of integrations is related to the fixed geometric data above the limiting kernel is believed to be a universal master kernel

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

dirichlet forms and diffusion processes for fermion random point fields abstract we construct the dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the euclidean space r d for which certain fermion random point fields are invariant

new two dimensional slope limiters for discontinuous galerkin methods on arbitrary meshes summary in this paper we introduce an extension of van leer s slope limiter for two dimensional discontinuous galerkin dg methods on arbitrary unstructured quadrangular or triangular grids the aim is to construct a non oscillatory shock capturing dg method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion unlike some splitting techniques that are limited to linear approximations on rectangular grids in this work the solution is approximated by means of piecewise quadratic functions the main idea of this new reconstructing and limiting technique follows a well known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters copyright 2004 john wiley sons ltd

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

riemann hilbert approach to multi time processes the airy and the pearcey cases we prove that matrix fredholm determinants related to multi time processes can be expressed in terms of determinants of integrable kernels a la its izergin korepin slavnov iiks and hence related to suitable riemann hilbert problems thus extending the known results for the single time case we focus on the airy and pearcey processes as an example of applications we re deduce a third order pde found by adler and van moerbeke for the two time airy process

airy kernel with two sets of parameters in directed percolation and random matrix theory we introduce a generalization of the extended airy kernel with two sets of real parameters we show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices

laminar separation bubble characteristics on an airfoil at low reynolds numbers an experimental investigation was conducted in order to document the structure and behavior of laminar separation bubbles at low reynolds numbers data of this type are necessary if the currently insufficient analytical and numerical models are to be improved the laminar separation bubble that forms on a naca 663 018 airfoil model was surveyed at chord reynolds numbers of 50 000 200 000 at angles of attack of 8 12 deg the effects of the various testing conditions on the separation bubble were isolated and the data were analyzed in relation to existing separation bubble correlations in order to test their low reynolds number applicability this analysis indicated that the chord reynolds number and the disturbance environment strongly influence the experimental pressure distributions these effects must be included in any analytic prediction technique applied to the low reynolds number flight regime

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

an o p3 hp version fem in two dimensions preconditioning and post processing abstract the bernstein polynomials have been known for over a century and are widely used in the spline literature computer aided geometric design and computer graphics however the realization that the bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development for instance it is known ainsworth et al 2011 that the bernstein basis can be exploited to compute all of the entries in the load vector in o p 3 operations even in the case of non linear problems on curvilinear elements for a degree p approximation moreover the element matrices can be assembled in o 1 operations per entry we show that properties of the bernstein polynomials can also be exploited to obtain o p 3 complexity procedures for all of the main components needed to implement a high order finite element code including computation of the residuals needed for an iterative solution method evaluating the action of a preconditioner for the global mass matrices and visualization and post processing of the resulting finite element approximations the construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p at a cost of o p 3 operations is one of the main contributions of the present work the preconditioner is based on an abstract additive schwarz method recently developed by the authors the preconditioner can be implemented at a cost of o p 3 operations by exploiting properties of the bernstein polynomials in particular we present an algorithm which allows one to invert the interior block of the element mass matrix in o p 3 operations numerical examples are provided to illustrate the applicability of the bernstein basis to challenging non linear reaction diffusion problems non linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

essential ingredients of a method for low reynolds number airfoils a calculation method for low reynolds number flows is described and appraised in terms of comparisons between calculations and measurements it comprises interaction between solutions of inviscid and boundary layer equations and involves the e n method to determine the location of the onset of transition with further interaction to insure that upstream effects of the turbulent flow region are correctly represented

fast simplicial quadrature based finite element operators using bernstein polynomials we derive low complexity matrix free finite element algorithms for simplicial bernstein polynomials on simplices our techniques based on a sparse representation of differentiation and special block structure in the matrices evaluating b form polynomials at warped gauss points apply to variable coefficient problems as well as constant coefficient ones thus extending our results in kirby numer math 2011 in press

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

gibbs cluster measures on configuration spaces the probability distribution gcl of a gibbs cluster point process in x rd with i i d random clusters attached to points of a gibbs configuration with distribution g is studied via the projection of an auxiliary gibbs measure g in the space of configurations x y x x where x x indicates a cluster center and y x nxn represents a corresponding cluster relative to x we show that the measure gcl is quasi invariant with respect to the group diff0 x of compactly supported diffeomorphisms of x and prove an integration by parts formula for gcl the associated equilibrium stochastic dynamics is then constructed using the method of dirichlet forms

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

analysis and geometry on configuration spaces abstract in this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds namely configuration spaces more precisely a differential geometry is introduced on the configuration space x over a riemannian manifold x this geometry is non flat even if x r d it is obtained as a natural lifting of the riemannian structure on x in particular a corresponding gradient divergence div and laplace beltrami operator h div gamma are constructed the associated volume elements i e all measures on x w r t which and div gamma become dual operators on l 2 x are identified as exactly the mixed poisson measures with mean measure equal to a multiple of the volume element dx on x in particular all these measures obey an integration by parts formula w r t vector fields on x the corresponding dirichlet forms e on l 2 x are therefore defined each is shown to be associated with a diffusion process which is thus the brownian motion on x and which is sub sequently identified as the usual independent infinite particle process on x the associated heat semigroup t t t 0 is calculated explicitly it is also proved that the diffusion process when started with is time ergodic or equivalently e gamma is irreducible or equivalently t t t 0 is ergodic if and only if is poisson measure z dx with intensity z dx for some z 0 furthermore it is shown that the laplace beltrami operator h div gamma gamma on l 2 x z dx is unitary equivalent to the second quantization of the laplacian x on x on the corresponding fock space n 0 l 2 x z dx n as another direct consequence of our results we obtain a representation of the lie algebra of compactly supported vector fields on x on poisson space finally generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on x are described in particular the case where the mixed poisson measures are replaced by gibbs measures of ruelle type on x

quasi invariance and integration by parts for determinantal and permanental processes abstract determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent their atoms exhibit mutual attraction of repulsion thus these processes are very far from the uncorrelated situation encountered in poisson models we establish a quasi invariance result we show that if atom locations are perturbed along a vector field the resulting process is still a determinantal respectively permanental process the law of which is absolutely continuous with respect to the original distribution based on this formula following bismut approach of malliavin calculus we then give an integration by parts formula

geometric decompositions and local bases for spaces of finite element differential forms abstract we study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions these spaces are generalizations of the classical finite element spaces for vector fields frequently referred to as raviart thomas brezzi douglas marini and nedelec spaces in the present paper we derive geometric decompositions of these spaces which lead directly to explicit local bases for them generalizing the bernstein basis for ordinary lagrange finite elements the approach applies to both families of finite element spaces for arbitrary polynomial degree arbitrary order of the differential forms and an arbitrary simplicial triangulation in any number of space dimensions a prominent role in the construction is played by the notion of a consistent family of extension operators which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis

limit process of stationary tasep near the characteristic line the totally asymmetric simple exclusion process tasep on input amssym bbb z with the bernoulli measure as an initial condition 0 1 is stationary it is known that along the characteristic line the current fluctuates at an order of t1 3 the limiting distribution has also been obtained explicitly in this paper we determine the limiting multipoint distribution of the current fluctuations moving away from the characteristics by the order t2 3 the main tool is the analysis of a related directed last percolation model we also discuss the process limit in tandem queues in equilibrium 2010 wiley periodicals inc

subcell flux limiting for high order bernstein finite element discretizations of scalar hyperbolic conservation laws abstract this work extends the concepts of algebraic flux correction and convex limiting to continuous high order bernstein finite element discretizations of scalar hyperbolic problems using an array of adjustable diffusive fluxes the standard galerkin approximation is transformed into a nonlinear high resolution scheme which has the compact sparsity pattern of the piecewise linear or multilinear subcell discretization the representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under cfl like conditions in contrast to predictor corrector approaches based on the flux corrected transport methodology the proposed flux limiting strategy is monolithic i e limited antidiffusive terms are incorporated into the well defined residual of a nonlinear semi discrete problem a stabilized high order galerkin discretization is recovered if no limiting is performed in the limited version the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors a formal proof of sparsity is provided for simplicial and box elements the involved element contributions can be calculated efficiently making use of matrix free algorithms and precomputed element matrices of the reference element numerical studies for q 2 discretizations of linear and nonlinear two dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions

analysis of airfoil leading edge separation bubbles a local inviscid viscous interaction technique was developed for the analysis of low speed airfoil leading edge transitional separation bubbles in this analysis an inverse boundary layer finite difference analysis is solved iteratively with a cauchy integral representation of the inviscid flow which is assumed to be a linear perturbation to a known global viscous airfoil analysis favorable comparisons with data indicate the overall validity of the present localized interaction approach in addition numerical tests were performed to test the sensitivity of the computed results to the mesh size limits on the cauchy integral and the location of the transition region

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

dimer pair correlations on the brick lattice using exact methods pair correlation functions are studied in the dimer model defined on a brick lattice at long distances these functions exhibit strongly anisotropic algebraic decay and near criticality the length scales diverge differently in the two principal directions the critical exponents are x 1 2 and y 1 these results are in agreement with deductions drawn from recent exact finite size scaling calculations we also interpret our results in the light of domain wall theories of commensurate incommensurate transitions and in particular we study the relation of the present model to the discrete version of the pokrovsky talapov model introduced by villain

recursive parameter free explicitly defined interpolation nodes for simplices a rule for constructing interpolation nodes for n th degree polynomials on the simplex is presented these nodes are simple to define recursively from families of 1d node sets such as the lobatto

statistics of layered zigzags a two dimensional generalization of tasep a novel discrete growth model in 2 1 dimensions is presented in three equivalent formulations i directed motion of zigzags on a cylinder ii interacting interlaced tasep layers and iii growing heap over 2d substrate with a restricted minimal local height gradient we demonstrate that the coarsegrained behavior of this model is described by the two dimensional kardar parisi zhang equation the coefficients of different terms in this hydrodynamic equation can be derived from the steady state flow density curve the so called fundamental diagram a conjecture concerning the analytical form of this flow density curve is presented and is verified numerically

transversal fluctuations for increasing subsequences on the plane consider a realization of a poisson process in 2 with intensity 1 and take a maximal up right path from the origin to n n consisting of line segments between the points where maximal means that it contains as many points as possible the number of points in such a path has fluctuations of order n where 1 3 bdj here we show that typical deviations of a maximal path from the diagonal x y is of order n with 2 3 this is consistent with the scaling identity 2 1 which is believed to hold in many random growth models

moment formulae for general point processes abstract the goal of this paper is to generalize most of the moment formulae obtained in 12 more precisely we consider a general point process and show that the quantities relevant to our problem are the so called papangelou intensities when the papangelou intensities of are well defined we show some general formulae to recover the moment of order n of the stochastic integral of the point process we will use these extended results to introduce a divergence operator and study a random transformation of the point process

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices