a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

a block preconditioner for non isothermal flow in porous media abstract in petroleum reservoir simulation the industry standard preconditioner the constrained pressure residual method cpr is a two stage process which involves solving a restricted pressure system with algebraic multigrid amg initially designed for isothermal models this approach is often used in the thermal case however it does not have a specific treatment of the additional energy conservation equation and temperature variable we seek to develop preconditioners which better capture thermal effects such as heat diffusion in order to study the effects of both pressure and temperature on fluid and heat flow we consider a model of non isothermal single phase flow through porous media for this model we develop a block preconditioner with an efficient schur complement approximation both the pressure block and the approximate schur complement are approximately inverted using an amg v cycle the resulting solver is scalable with respect to problem size and parallelization

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

a solvable mixed charge ensemble on the line global results we consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2 1 in the presence of an external field with the total charge fixed and the system held at temperature corresponding to 1 it is proved that the particles form a pfaffian point process when the external field is quadratic the harmonic oscillator potential we produce the explicit family of skew orthogonal polynomials necessary to simplify the related matrix kernels in this setting a variety of limit theorems are proved on the distribution of the number as well as the spatial density of each species of particle as the total charge increases to infinity connections to ginibre s real ensemble of random matrix theory are highlighted throughout

small sample properties of the maximum likelihood estimator in the first order moving average model summary in this paper we discuss finite sample properties of the maximum likelihood estimator 0 in the first order moving average model we give a theoretical explanation for the concentration of 0 values at the invertibility boundary we derive the exact distribution of 0 for sample size n 2 which is found to be of mixed type for general n we give approximations for pr 0 1

solver composition across the pde linear algebra barrier the efficient solution of discretizations of coupled systems of partial differential equations pdes is at the core of much of numerical simulation significant effort has been expended on scalable algorithms to precondition krylov iterations for the linear systems that arise with few exceptions the reported numerical implementation of such solution strategies is specific to a particular model setup and intimately ties the solver strategy to the discretization and pde especially when the preconditioner requires auxiliary operators in this paper we present recent improvements in the firedrake finite element library that allow for straightforward development of the building blocks of extensible composable preconditioners that decouple the solver from the model formulation our implementation extends the algebraic composability of linear solvers offered by the petsc library by augmenting operators and hence preconditioners with the ability to provide any necessary auxiliary operators rather than s

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

general eigenvalue correlations for the real ginibre ensemble we rederive in a simplified version the lehmann sommers eigenvalue distribution for the gaussian ensemble of asymmetric real matrices invariant under real orthogonal transformations as a basis for a detailed derivation of a pfaffian generating functional for n point densities this produces a simple free fermion diagram expansion for the correlations leading to quaternion determinants in each order n all will explicitly be given with the help of a very simple symplectic kernel for even dimension n the kernel is valid for both complex and real eigenvalues and describes a deep connection between both a slight modification by an artificial additional grassmannian also solves the more complicated odd n case as illustration we present some numerical results in the space of complex eigenvalue n tuples

a combinatorial proof of the giambelli identity for schur functions the giambelli identity provides a formula for expressing an arbitrary schur function of shape as a determinant of schur functions of hook shapes l 1k we give the first complete combinatorial proof of the giambelli identity we show how to derive various hook formulas from the giambelli identity and show how to extend our methods to derive extensions of the giambelli identity and the hook formula for the number of standard tableaux to certain skew shapes

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

classical skew orthogonal polynomials and random matrices skew orthogonal polynomials arise in the calculation of the n point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry in particular the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials in the case that the eigenvalue probability density function involves a classical weight function explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials and the structure is used to give a closed form expression for the sum this theory treates all classical cases on an equal footing giving formulas applicable at once to the hermite laguerre and jacobi cases

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

maximum likelihood estimation of regression models with stochastic trend components abstract this article is concerned with the estimation of a regression model with a stochastic trend component it is shown that the probability of estimating the trend to be deterministic is very sensitive to the type of likelihood function used as the basis of inference

a generalized plasma and interpolation between classical random matrix ensembles the eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log gas at the special couplings 1 2 and 4 it has been known for some time that there is an exactly solvable two component log potential plasma which interpolates between the 1 and 4 circular ensemble and an exactly solvable two component generalized plasma which interpolates between 2 and 4 circular ensemble we extend known exact results relating to the latter for the free energy and one and two point correlations by giving the general k1 k2 point correlation function in a pfaffian form crucial to our working is an identity which expresses the vandermonde determinant in terms of a pfaffian the exact evaluation of the general correlation is used to exhibit a perfect screening sum rule

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

a dual of macmahon s theorem on plane partitions a classical theorem of macmahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula in this paper we present a counterpart of this formula corresponding to the exterior of a concave hexagon obtained by turning 120 after drawing each side macmahon s hexagon is obtained by turning 60 after each step

effective laguerre asymptotics it is known that the generalized laguerre polynomials can enjoy subexponential growth for large primary index in particular for certain fixed parameter pairs a z one has the large n asymptotic behavior l n a z sim c a z n a 2 1 4 e 2 sqrt nz we introduce a computationally motivated contour integral that allows efficient numerical laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior we present a fast algorithm for symbolic generation of the rather formidable expansion coefficients along the way we address the difficult problem of establishing effective i e rigorous and explicit error bounds on the general expansion a primary tool for these developments is an exp arc method giving a natural bridge between converging series and effective asymptotics

the ginibre ensemble of real random matrices and its scaling limits we give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the pfaffian of an antisymmetric matrix formed from a 2 2 matrix kernel associated to the ensemble we apply this result to the real ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large

empirical likelihood confidence regions in time series models summary this paper develops empirical likelihood in time series models by application of whittle s estimation method one obtains an m estimator of the parameter of a stationary time series from approximately independent observations the periodogram ordinates this estimator is used to obtain an empirical likelihood ratio which is asymptotically distributed as x2 and which can be used to construct confidence regions a procedure for the bartlett correction is also proposed finally small sample properties of the empirical likelihood confidence regions are explored through a simulation

vicious walkers friendly walkers and young tableaux iii between two walls we derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls as well as corresponding results for the analogous problem of friendly walkers our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

iterative solution methods for modeling multiphase flow in porous media fully implicitly we discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell centered finite difference discretization of two and three phase porous media flow every outer nonlinear newton iteration requires solution of a nonsymmetric jacobian linear system two major types of preconditioners supercoarsening multigrid scmg and two stage are developed for the gmres iteration applied to the solution of the jacobian system the scmg reduces the three dimensional system to two dimensions using a vertical aggregation followed by a two dimensional multigrid the two stage preconditioners are based on decoupling the system into a pressure and concentration equations several pressure preconditioners of different types are described extensive numerical results are presented using the integrated parallel reservoir simulator ipars and indicate that these methods have low arithmetical complexity per iteration and good convergence rates

algebraic multigrid preconditioners for multiphase flow in porous media multiphase flow is a critical process in a wide range of applications including carbon sequestration contaminant remediation and groundwater management typically this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase e g oil water along with constitutive laws for the relationship of phase velocity to phase pressure in this study we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time implicit treatment of models of multiphase flow we explore the performance of several preconditioners based on algebraic multigrid amg for solving the linearized problem including black box amg applied directly to the system a new version of constrained pressure residual multigrid cpr amg preconditioning and a new preconditioner derived using an approximate schur complement arising from the block factorization of the jacobian we show that th