the uniqueness of the asymmetric jet flow abstract in the previous significant works by alt et al 1982 2 and alt et al 1985 6 the existence of the two dimensional asymmetric incompressible jet flow and compressible subsonic jet flow were established however the uniqueness of the asymmetric jet flow has remained an unsolved open problem in this paper we will establish the uniqueness of asymmetric jet flow not only for incompressible case but also for compressible subsonic case furthermore as a byproduct some properties of the jet flow are obtained such as the location estimate and asymptotic deflection angle estimate of the jet in far field finally a comparison theorem will be introduced to compressible subsonic jet flow

compressible subsonic jet flows issuing from a nozzle of arbitrary cross section abstract we are concerned with the well posedness theory of two dimensional compressible subsonic jet flow issuing from a semi infinitely long nozzle of arbitrary cross section given any atmospheric pressure p 0 we show that there exists a critical mass flux m c r depending on p 0 and such that if the incoming mass flux m 0 is less than the critical value then there exists a unique smooth subsonic jet flow issuing from the given nozzle the jet boundary is a free streamline which initiates from the end point of the nozzle smoothly and extends to the infinity one of the key observations in this paper is that the restriction of the incoming mass flux guarantees completely the subsonicity of the compressible jet in the whole flow field which coincides with the observation on the compressible subsonic flows in an infinitely long nozzle without free boundary in 8

a posteriori analysis and adaptive error control for multiscale operator decomposition solution of elliptic systems i triangular systems this paper we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems the goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition our approach to error estimation is based on a well known a posteriori analysis involving variational analysis residuals and the generalized green s function our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition in part i of this paper we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components we also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition in part ii of this paper we address issues related to the iterative solution of a fully coupled nonlinear system

quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

influence functions and goal oriented error estimation for finite element analysis of shell structures in this paper we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures where the influence function can be for any linear quantity of engineering interest we then design some goal oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over estimation these error measures include the error due to the approximation in the geometry of the shell structure in the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero although our procedures are general and can be applied to any shell formulation we focus on mitc finite element shell discretizations in our numerical results influence functions are shown for some shell test problems and the proposed goal oriented error estimation procedure shows good effectivity indices copyright 2005 john wiley sons ltd

stability of supersonic contact discontinuity for two dimensional steady compressible euler flows in a finite nozzle abstract in this paper we study the stability of supersonic contact discontinuity for the two dimensional steady compressible euler flows in a finitely long nozzle of varying cross sections we formulate the problem as an initial boundary value problem with the contact discontinuity as a free boundary to deal with the free boundary value problem we employ the lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem we develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain finally by using the inverse lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

generalizing global error estimation for ordinary differential equations by using coupled time stepping methods abstract this study introduces new time stepping strategies with built in global error estimators the new methods propagate the defect along with the numerical solution much like solving for the correction or zadunaisky s procedure however the proposed approach allows for overlapped internal computations and therefore represents a generalization of the classical numerical schemes for solving differential equations with global error estimation the resulting algorithms can be effectively represented as general linear methods several explicit self starting schemes akin to runge kutta methods with global error estimation are introduced and the theoretical considerations are illustrated in several examples

goal oriented adaptivity for gmsfem in this paper we develop two goal oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework in this methodology one seeks to determine the number of multiscale basis functions adaptively for each coarse region to efficiently reduce the error in the goal functional our first error estimator uses a residual based strategy where local indicators on each coarse neighborhood are the product of local indicators for the primal and dual problems respectively in the second approach viewed as the multiscale extension of the dual weighted residual method dwr the error indicators are computed as the pairing of the local h 1 residual of the primal problem weighed by a projection into the primal space of the h 0 1 dual solution from an enriched space over each coarse neighborhood in both of these strategies the goal oriented indicators are then used in place of a standard residual based indicator to mark coarse neighborhoods of the mesh for further enrichment in the form of additional multiscale basis functions the method is demonstrated on high contrast problems with heterogeneous multiscale coefficients and is seen to outperform the standard residual based strategy with respect to efficient reduction of error in the goal function

two phase fluids in collision of incompressible inviscid fluids effluxing from two nozzles abstract this paper is devoted to the mathematical theory of two dimensional injection of incompressible irrotational and inviscid fluids issuing from two infinitely long nozzles into a free stream in general there is a free interface with constant jump of the bernoulli constant on it which is different and more difficult than the previous related works physically it is called the collision fluid the main result in this paper is that for given two co axis symmetric infinitely long nozzles imposing the incoming mass fluxes in the two nozzles there exists a unique piecewise smooth collision fluid such that the free interface of the collision fluid is a c 1 curve and the pressure is continuous across the interface as byproducts the asymptotic behaviors the positivity of the vertical velocity monotone relationship between the location of the interface and the incoming mass fluxes are also established

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

a variational principle for a non integrable model we show the existence of a variational principle for graph homomorphisms from mathbb z m to a d regular tree the technique is based on a discrete kirszbraun theorem and a concentration inequality obtained through the dynamics of the model as another consequence of the concentration inequality we also obtain the existence of a continuum of translation invariant ergodic gradient gibbs measures for graph homomorphisms from mathbb z m to a regular tree the method is sufficiently robust such that it could be applied to other discrete models with a quite general target graphs

ribbon tilings and multidimensional height functions we fix n and say a square in the two dimensional grid indexed by x y has color c if x y c mod n a ribbon tile of order n is a connected polyomino containing exactly one square of each color we show that the set of order n ribbon tilings of a simply connected region r is in one to one correspondence with a set of height functions from the vertices of r to z n satisfying certain difference restrictions it is also in one to one correspondence with the set of acyclic orientations of a certain partially oriented graph using these facts we describe a linear in the area of r algorithm for determining whether r can be tiled with ribbon tiles of order n and producing such a tiling when one exists we also resolve a conjecture of pak by showing that any pair of order n ribbon tilings of r can be connected by a sequence of local replacement moves some of our results are generalizations of known results for order 2 ribbon tilings a k a domino tilings we also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems

arctic circles domino tilings and square young tableaux the arctic circle theorem of jockusch propp and shor asserts that uniformly random domino tilings of an aztec diamond of high order are frozen with asymptotically high probability outside the arctic circle inscribed within the diamond a similar arctic circle phenomenon has been observed in the limiting behavior of random square young tableaux in this paper we show that random domino tilings of the aztec diamond are asymptotically related to random square young tableaux in a more refined sense that looks also at the behavior inside the arctic circle this is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the aztec diamond that uses the large deviation techniques developed for the square young tableaux problem in a previous paper by pittel and the author the solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square young tableaux by pittel and the author the analytic techniques used to solve the variational problem provide a systematic guess free approach for solving problems of this type which have appeared in a number of related combinatorial probability models

two dimensional cavity flow in an infinitely long channel with non zero vorticity abstract the main object of this paper is to investigate the well posedness theory of the incompressible inviscid cavity flow in an infinitely long channel the flow is governed by two dimensional incompressible steady euler system the main results read that given a mass flux and a constant vorticity in the inlet of the channel firstly we establish the existence and the uniqueness of the incompressible cavity flow in an infinitely long symmetric channel which contains a smooth free surface detaching at the boundary point of the obstacle secondly some fundamental properties such as the asymptotic behaviors of the cavity flow and the free boundary in the upstream and downstream and the positivity of the horizontal velocity are also obtained finally we show that there does not exist a finite cavity or a cusped cavity in the infinitely long nozzle which gives a positive answer to the conjecture by h villat in 1913 on the non existence of a symmetric finite cusped cavity behind an obstacle

local statistics of realizable vertex models we study planar vertex models which are probability measures on edge subsets of a planar graph satisfying certain constraints at each vertex examples including the dimer model and 1 2 model which we will define we express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding fisher graph with the help of a generalized holographic algorithm using an n n torus to approximate the periodic infinite graph we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi periodic graph as an example we simulate the 1 2 model by the technique of glauber dynamics

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

tacnode gue minor processes and double aztec diamonds we study determinantal point processes arising in random domino tilings of a double aztec diamond a region consisting of two overlapping aztec diamonds at a turning point in a single aztec diamond where the disordered region touches the boundary the natural limiting process is the gue minor process increasing the size of a double aztec diamond while keeping the overlap between the two aztec diamonds finite we obtain a new determinantal point process which we call the tacnode gue minor process this process can be thought of as two colliding gue minor processes as part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode gue minor process we find the inverse kasteleyn matrix for the dimer model version of the double aztec diamond

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles abstract in this paper we establish existence of global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles by combining variational method various elliptic estimates and a compensated compactness method more precisely it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value moreover uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields and flow angles for axially symmetric flows are uniformly bounded away from 2 finally when the incoming mass flux tends to the critical value subsonic sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework

a posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

loop statistics in the toroidal honeycomb dimer model the dimer model on a graph embedded in the torus can be interpreted as a collection of random self avoiding loops in this paper we consider the uniform toroidal honeycomb dimer model we prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed the winding number of the collection of loops converges in law to a two dimensional discrete gaussian distribution this is known to physicists in more generality from their analysis of toroidal two dimensional critical loop models and their mapping to the massless free field on the torus this paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model

fast and reliable methods for determining the evolution of uncertain parameters in differential equations a very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model the monte carlo method is a justifiably popular tool for analyzing such effects it does however require many evaluations of the operator and it is difficult to extract precise information about the accuracy of any particular result in this paper we borrow techniques from a posteriori error analysis for finite element methods to compute information about the derivative of an operator with respect to its parameters these techniques employ the generalized green s function to describe how variation propagates into the solution around localized points in the parameter space we show how this derivative information can be used either to create a higher order method or produce an error estimate for information computed from a given representation in the latter case this provides the basis for adaptive sampling according to the variation in the output values both the higher order method and the adaptive sampling method are generally orders of magnitude faster than monte carlo methods in the case that the parameter space is not too high dimensional

variational multiscale a posteriori error estimation for systems application to linear elasticity abstract this paper presents an explicit residual based a posteriori error estimator for linear elasticity the method is derived from the variational multiscale theory where the subgrid scales and the constants of the method are calculated explicitly from fine scale green s functions several numerical examples are presented showing satisfactory local and global efficiencies when the h 1 seminorm is employed the formulation which is applied to plane stress problems can be easily extended to other elasticity problems

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

configurational entropy of codimension one tilings and directed membranes the calculation of random tiling configurational entropy amounts to an enumeration of partitions a geometrical description of the configuration space is given in terms of integral points in a high dimensional space and the entropy is deduced from the integral volume of a convex polytope in some cases the latter volume can be expressed in a compact multiplicative formula and in all cases in terms of binomial series the origin of which is given a geometrical meaning our results mainly concern codimension one tilings but can also be extended to higher codimension tilings we also discuss the link between freeboundary and fixed boundary condition problems

a combined approach for goal oriented error estimation and adaptivity using operator customized finite element wavelets we describe how wavelets constructed out of finite element interpolation functions provide a simple and convenient mechanism for both goal oriented error estimation and adaptivity in finite element analysis this is done by posing an adaptive refinement problem as one of compactly representing a signal the solution to the governing partial differential equation in a multiresolution basis to compress the solution in an efficient manner we first approximately compute the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh the estimation step and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to a functional of interest the adaptation step in this sense therefore the proposed approach is unified since unlike many contemporary error estimation and adaptive refinement methods the basis functions used for error estimation are the same as those used for adaptive refinement we illustrate the application of the proposed technique for goal oriented error estimation and adaptivity for second and fourth order linear elliptic pdes and demonstrate its advantages over existing methods copyright 2005 john wiley sons ltd

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

generalized green s functions and the effective domain of influence one well known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized green s function the generalized green s function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability including any decay of influence characteristic of elliptic problems we show that consideration of the generalized green s function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and or to compute quantities of interest that involve globally supported information such as average values and norms in the latter case we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions by treating each computation of a quantity of interest independently the maximum number of elements required to achieve the desired accuracy can be decreased significantly

fast methods for determining the evolution of uncertain parameters in reaction diffusion equations abstract an important problem in mathematical modeling in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error in measured parameter values or may arise as part of a sensitivity analysis of the model the monte carlo method is a widely used tool for determining such effects it employs random sampling of the input space in order to produce a pointwise representation of the output it is a robust and easily implemented tool with relatively low dependence on the number of parameters unfortunately it generally requires sampling the operator very many times at a significant cost especially when the model is expensive to evaluate moreover standard analysis provides only asymptotic or distributional information about the error computed from a particular realization in this paper we present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a reaction diffusion equation on the solution the approach is based on techniques borrowed from a posteriori error analysis for finite element methods the generalized green s function solving the adjoint problem is used to efficiently compute the gradient of a quantity of interest with respect to parameters at sample points in the parameter space this derivative information is used in turn to produce an error estimate for the information computed from the sample points the error estimate provides the basis for both deterministic and probabilistic adaptive sampling algorithms the deterministic adaptive sampling method can be orders of magnitude faster than monte carlo sampling in case of a moderate number of parameters the gradient can also be used to compute useful information that cannot be obtained easily from a monte carlo sample for example the adaptive algorithm yields a natural dimensional reduction in the parameter space where applicable after describing the new methodology in detail we apply the new method to analyze the parameter sensitivity of a predator prey model with a holling ii functional response that has six parameters

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

a posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid solid heat transfer we analyze a multiscale operator decomposition finite element method for a conjugate heat transfer problem consisting of a fluid and a solid coupled through a common boundary we derive accurate a posteriori error estimates that account for all sources of error and in particular the transfer of error between fluid and solid domains we use these estimates to guide adaptive mesh refinement in addition we provide compelling numerical evidence that the order of convergence of the operator decomposition method is limited by the accuracy of the transferred gradient information and adapt a so called boundary flux recovery method developed for elliptic problems in order to regain the optimal order of accuracy in an efficient manner in an appendix we provide an argument that explains the numerical results provided sufficient smoothness is assumed

global subsonic euler flows in an infinitely long axisymmetric nozzle abstract in this paper we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle the flow is governed by the steady euler equations and has boundary conditions on the nozzle walls existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle when the variation of bernoulli s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value the results give a strictly mathematical proof to the assertion in bers 1958 2 there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle provided that the incoming mass flux is less than the critical value the existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables with the assumptions on the nozzle in the far fields the asymptotic behavior can be derived by a blow up argument for the infinitely long nozzle finally we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux

on two dimensional sonic subsonic flow a compensated compactness framework is established for sonic subsonic approximate solutions to the two dimensional euler equations for steady irrotational flows that may contain stagnation points only crude estimates are required for establishing compactness it follows that the set of subsonic irrotational solutions to the euler equations is compact thus flows with sonic points over an obstacle such as an airfoil may be realized as limits of sequences of strictly subsonic flows furthermore sonic subsonic flows may be constructed from approximate solutions the compactness framework is then extended to self similar solutions of the euler equations for unsteady irrotational flows

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

subsonic sonic limit of approximate solutions to multidimensional steady euler equations a compactness framework is established for approximate solutions to subsonic sonic flows governed by the steady full euler equations for compressible fluids in arbitrary dimension the existing compactness frameworks for the two dimensional irrotational case do not directly apply for the steady full euler equations in higher dimensions the new compactness framework we develop applies for both non homentropic and rotational flows one of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity along with the bernoulli law and the entropy relation through a more delicate analysis on the phase space as direct applications we establish two existence theorems for multidimensional subsonic sonic full euler flows through infinitely long nozzles

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

worst case multi objective error estimation and adaptivity this paper introduces a new computational methodology for determining a posteriori multi objective error estimates for finite element approximations and for constructing corresponding quasi optimal adaptive refinements of finite element spaces as opposed to the classical goal oriented approaches which consider only a single objective functional the presented methodology applies to general closed convex subsets of the dual space and constructs a worst case error estimate of the finite element approximation error this worst case multi objective error estimate conforms to a dual weighted residual in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error we regard both standard approximation errors and data incompatibility errors associated with incompatibility of boundary data with the trace of the finite element space numerical experiments are presented to demonstrate the efficacy of applying the proposed worst case multi objective error estimate in adaptive refinement procedures

asymptotically exact functional error estimators based on superconvergent gradient recovery the use of dual adjoint problems for approximating functionals of solutions of pdes with great accuracy or to merely drive a goal oriented adaptive refinement scheme has become well accepted and it continues to be an active area of research the traditional approach involves dual residual weighting drw in this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact the first is of drw type and is derived for meshes in which most triangles satisfy an inlinemediaobject not available see fulltext approximate parallelogram property the second functional estimator involves dual error estimate weighting dew using any superconvergent gradient recovery technique for the primal and dual solutions several experiments are done which demonstrate the asymptotic exactness of a dew estimator which uses a gradient recovery scheme proposed by bank and xu and the effectiveness of refinement done with respect to the corresponding local error indicators

flip dynamics in three dimensional random tilings we study single flip dynamics in sets of three dimensional rhombus tilings with fixed polyhedral boundaries this dynamics is likely to be slowed down by so called cycles such structures arise when tilings are encoded via the partition on tiling method and are susceptible to break connectivity by flips or at least ergodicity because they locally suppress a significant amount of flip degrees of freedom we first address the so far open question of the connectivity of tiling sets by elementary flips we prove exactly that sets of tilings of codimension 1 and 2 are connected for any dimension and tiling size for higher codimension tilings of dimension 3 the answer depends on the precise choice of the edge orientations which is a non trivial issue in most cases we can prove connectivity despite the existence of cycles in the few remaining cases among which one is the icosahedral symmetry the question remains open we also study numerically flip assisted diffusion to explore the possible effects of the previously mentioned cycles cycles do not seem to slow down significantly the dynamics at least as far as self diffusion is concerned

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

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

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

ribbon tile invariants let t be a finite set of tiles and b a set of regions f tileable by t we introduce a tile counting group g t b as a group of all linear relations for the number of times each tile t c t can occur in a tiling of a region r c b we compute the tile counting group for a large set of ribbon tiles also known as rim hooks in a context of representation theory of the symmetric group the tile counting group is presented by its set of generators which consists of certain new tile invariants in a special case these invariants generalize the conway lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group the heart of the proof is the known bijection between rim hook tableaux and certain standard skew young tableaux we also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem

steady subsonic ideal flows through an infinitely long nozzle with large vorticity in this paper the existence uniqueness and far field behavior of a class of subsonic flows with large vorticity for the steady euler equations in infinitely long nozzles are established more precisely for any given convex horizontal velocity of incoming flow in the upstream there exists a critical value mcr if the mass flux is larger than mcr then there exists a unique smooth subsonic euler flow through the infinitely long nozzle this well posedness result is proved by a new observation for the method developed in xie and xin siam j math anal 42 751 784 2010 to deal with the euler system furthermore the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

s4 symmetry on the checkerboard potts model the large q expansion of the partition function z of the checkerboard q state potts model with a magnetic field is given up to sixth order in q 12 up to this order z is shown to exhibit an unexpected s4 symmetry using various arguments exact results expansion etc the authors suggest the general validity of this new symmetry property of the considered model an exact expression for the magnetisation jump is also proposed and checked on the large q expansion

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

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

on incompressible oblique impinging jet flows abstract the object of this work is to investigate the fluid mechanics of oblique impinging jet flows and to this end some existence and nonexistence results are initiated first we established the existence of incompressible oblique impinging jet plane flows with two asymptotic directions more precisely given a two dimensional semi infinitely long nozzle and a wall behind the nozzle for any given mass flux in the inlet of the nozzle then there exists a smooth incompressible oblique impinging jet flow with two asymptotic directions the impinging jet develops two free streamlines which initiate smoothly at the endpoints of the semi infinitely long nozzle and the speed on free streamlines remains a constant which can be determined by the impinging jet flow itself the asymptotic behaviors of the oblique impinging jet flows at the far fields the position of the stagnation points convexity of the free boundaries and other properties are also considered on another side it is showed in this paper that there does not exist an oblique impinging jet flow with one asymptotic direction generally

a new goal oriented formulation of the finite element method abstract in this paper we introduce analyze and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary value problem the objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal oriented adaptivity which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement in the current approach we first solve the adjoint problem then use the adjoint information as a minimization constraint for the primal problem as a result the constrained finite element solution is enhanced with respect to the quantities of interest while maintaining near optimality in energy norm we describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples

tiling with dominoes pictures of plane figures without holes graphic basis of the thurston algorithm parallelization uniqueness and decomposition resume le desormais classique algorithme de thurston 1990 pour le pavage des figures planes finies et sans trous par des dominos peut etre retrouve par la condition de hall sur les couplages dans les graphes bipartis et la consideration de plus courts chemins dans un graphe associe a la figure a paver ce point de vue nouveau permet une parallelisation en temps o log n avec n 3 processeurs sur un modele pram c est a dire de facon efficace classe nc ce meme algorithme resout la question de l unicite d un pavage et de facon plus generale donne une decomposition canonique de la figure

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

low mach number limit of steady flows through infinite multidimensional largely open nozzles abstract in this paper we justify of the low mach number limit of the steady irrotational euler flow through infinite multidimensional largely open nozzles firstly the existence and uniqueness of the incompressible flow are proved then the uniform estimates on the compressibility parameter e which is singular for the flows are established via a variational approach based on the compressible incompressible difference functions the limit is in the holder space and is unique moreover the convergence rate is of order e 2 including the pressure and for the compressible case the extra force influences the asymptotic behaver to the open end while the effect vanishes in the limiting process from compressible flows to the incompressible ones also for the compressible flows with general conservative forces the well posedness of subsonic solution are given which leads to the existence of subsonic sonic limit solutions

critical resonance in the non intersecting lattice path model we study the phase transition in the honeycomb dimer model equivalently monotone non intersecting lattice path model at the critical point the system has a strong long range dependence in particular periodic boundary conditions give rise to a resonance phenomenon where the partition function and other properties of the system depend sensitively on the shape of the domain

a posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem we consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary we derive accurate a posteriori error estimates that account for the transfer of error between components of the operator decomposition method as well as the differences between the adjoints of the full problem and the discrete iterative system we use these estimates to guide adaptive mesh refinement in addition we address a loss of order of convergence that results from the decomposition and show that the order of convergence is limited by the accuracy of the transferred gradient information we employ a boundary flux recovery method to regain the expected order of accuracy in an efficient manner

convergence of an adaptive finite element method for controlling local energy errors a number of works concerning rigorous convergence theory for adaptive finite element methods afems for controlling global energy errors have appeared in recent years however many practical situations demand afems designed to efficiently compute quantities which depend on the unknown solution only on some subset of the overall computational domain in this work we prove convergence of an afem for controlling local energy errors the first step in our convergence proof is the construction of novel a posteriori error estimates for controlling a weighted local energy error this weighted local energy notion admits versions of standard ingredients for proving convergence of afems such as quasi orthogonality and error contraction but modulo pollution terms which use weaker norms to measure effects of global solution properties on the local energy error we then prove several convergence results for afems based on various marking strategies including a contraction result in the case of convex polyhedral domains

tile invariants new horizons let t be a finite set of tiles the group of invariants g t introduced by pak trans ams 352 2000 5525 is a group of linear relations between the number of copies of tiles in tilings of the same region we survey known results about g the height function approach the local move property various applications and special cases

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

a partial order on the regions of ⁿ dissected by hyperplanes we study a partial order on the regions of r dissected by hyperplanes this includes a computation of the mobius function and in some cases of the homotopy type applications are presented to zonotopes the weak bruhat order on weyl groups and acyclic orientations of graphs 0 introduction let cc h h2 hk be a set of hyperplanes in rw then the components of rw uheh form a set i of open n cells we will call regions traditionally i has been studied in terms of enumeration for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in c for a thorough discussion of this problem see zaslavsky s monograph zi and the references therein see also os and te in this paper we will study i from a different perspective our concern will be the way in which one can move from one region to another by walking through one hyperplane at a time to analyze this situation we define a partial order on j relative to a base region where the cover relations correspond to walking through a hyperplane and moving further away from the base the paper is organized as follows in 1 we define the partial order on rfl and begin to study its structure this culminates in theorem 1 11 where an explicit computation of its mbbius function is performed in 11 we examine the special case where all the hyperplanes go through the origin in certain cases we are able to compute the homotopy type of these posets the next three sections are devoted to applications 111 applies our work to a partial order on a zonotope iv deals with the weak bruhat order of a weyl group as a special case of our posets as a corollary we show that the homotopy type of the weak bruhat order is that of a sphere finally v applies our results to a partial order on the acyclic orientations of a graph some of the relationships used in iii iv and v appear in an enumerative context in the excellent article of greene and zaslavsky gz most of our poset notation is standard and can be found in a when more than one partial order is being used we will sometimes use subscripts to indicate where the relation is i e x py means that x is less than or equal to y in the poset p by l p we mean the simplicial complex of chains totally ordered sets in p we will think of p as its geometric realization by p we mean the order dual of p i e received by the editors july 13 1982 and in revised form may 15 1983 1980 mathematics subject classification primary 06a10 51m20 1984 american mathematical society 0002 9947 84 1 00 25 per page

subsonic flows in a multi dimensional nozzle in this paper we study the global subsonic irrotational flows in a multi dimensional n 2 infinitely long nozzle with variable cross sections the flow is described by the inviscid potential equation which is a second order quasilinear elliptic equation when the flow is subsonic first we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension with sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow then we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely provided that the incoming mass flux is less than the critical value this gives a positive answer to the problem of bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension bers in surveys in applied mathematics vol 3 wiley new york 1958 finally under suitable asymptotic assumptions of the nozzle we obtain the asymptotic behavior of the subsonic flow in far fields by means of a blow up argument the main ingredients of our analysis are methods of calculus of variations the moser iteration techniques for the potential equation and a blow up argument for infinitely long nozzles

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

enhancing least squares finite element methods through a quantity of interest in this paper we introduce an approach that augments least squares finite element formulations with user specified quantities of interest the method incorporates the quantity of interest into the least squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity of interest we establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions central to the approach is that it offers an element level estimate of the error in the quantity of interest as a result we introduce an adaptive approach that yields efficient adaptively refined approximations several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology notably the results show that the new approach is effective at improving the accuracy per total computational cost

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

quasi optimality of adaptive finite element methods for controlling local energy errors a rich theory demonstrating convergence and optimality of adaptive finite element methods afem has been developed in recent years in this work we prove optimality of afem which are designed to control local energy errors in elliptic partial differential equations because errors propagate globally in fem controlling local errors requires controlling both local energy solution properties and global error contributions pollution errors which may be measured in a weaker norm such as the l 2 l2 norm we define adaptive methods which control both of these error components and prove that they converge with the best possible rate over all possible refinements of the initial mesh these results are valid for poisson s problem on convex polyhedral domains in arbitrary space dimension our theory establishes afem optimality for several adaptive marking strategies which rigorously control pollution effects we also present numerical examples that illustrate our theory and confirm that local energy afem without pollution control can fail to yield optimal meshes